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Theorem List for Metamath Proof Explorer - 19101-19200   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremchpdmatlem1 19101 Lemma 1 for chpdmat 19104. (Contributed by AV, 18-Aug-2019.)
CharPlyMat        Poly1       Mat        algSc              var1              mulGrp              Mat                             matToPolyMat

Theoremchpdmatlem2 19102 Lemma 2 for chpdmat 19104. (Contributed by AV, 18-Aug-2019.)
CharPlyMat        Poly1       Mat        algSc              var1              mulGrp              Mat                             matToPolyMat

Theoremchpdmatlem3 19103 Lemma 3 for chpdmat 19104. (Contributed by AV, 18-Aug-2019.)
CharPlyMat        Poly1       Mat        algSc              var1              mulGrp              Mat                             matToPolyMat

Theoremchpdmat 19104* The characteristic polynomial of a diagonal matrix. (Contributed by AV, 18-Aug-2019.) (Proof shortened by AV, 21-Nov-2019.)
CharPlyMat        Poly1       Mat        algSc              var1              mulGrp              g

Theoremchpscmat 19105* The characteristic polynomial of a (nonempty!) scalar matrix. (Contributed by AV, 21-Aug-2019.)
CharPlyMat        Poly1       Mat        var1       mulGrp       .g              algSc

Theoremchpscmat0 19106* The characteristic polynomial of a (nonempty!) scalar matrix, expressed with its diagonal element. (Contributed by AV, 21-Aug-2019.)
CharPlyMat        Poly1       Mat        var1       mulGrp       .g              algSc

Theoremchpscmatgsumbin 19107* The characteristic polynomial of a (nonempty!) scalar matrix, expressed as finite group sum of binomials. (Contributed by AV, 2-Sep-2019.)
CharPlyMat        Poly1       Mat        var1       mulGrp       .g              algSc              .g       mulGrp       .g                     g

Theoremchpscmatgsummon 19108* The characteristic polynomial of a (nonempty!) scalar matrix, expressed as finite group sum of scaled monomials. (Contributed by AV, 2-Sep-2019.)
CharPlyMat        Poly1       Mat        var1       mulGrp       .g              algSc              .g       mulGrp       .g                     .g       g

Theoremchp0mat 19109 The characteristic polynomial of the zero matrix. (Contributed by AV, 18-Aug-2019.)
CharPlyMat        Poly1       Mat        var1       mulGrp       .g

Theoremchpidmat 19110 The characteristic polynomial of the identity matrix. (Contributed by AV, 19-Aug-2019.)
CharPlyMat        Poly1       Mat        var1       mulGrp       .g              algSc

Theoremchmaidscmat 19111 The characteristic polynomial of a matrix multiplied with the identity matrix is a scalar matrix. (Contributed by AV, 30-Oct-2019.) (Revised by AV, 19-Dec-2019.)
Mat               CharPlyMat        Poly1              Mat                                    ScMat

11.5.2  The characteristic factor function G

In this subsection the function is discussed. This function is involved in the representation of the product of the characteristic matrix of a given matrix and its adjunct as an infinite sum, see cpmadugsum 19141. Therefore, this function is called "characteristic factor function" (in short "chfacf") in the following. It plays an important role in the proof of the Cayley-Hamilton theorem, see cayhamlem1 19129, cayhamlem3 19150 and cayhamlem4 19151.

Theoremfvmptnn04if 19112* The function values of a mapping from the nonnegative integers with four distinct cases. (Contributed by AV, 10-Nov-2019.)

Theoremfvmptnn04ifa 19113* The function value of a mapping from the nonnegative integers with four distinct cases for the first case. (Contributed by AV, 10-Nov-2019.)

Theoremfvmptnn04ifb 19114* The function value of a mapping from the nonnegative integers with four distinct cases for the second case. (Contributed by AV, 10-Nov-2019.)

Theoremfvmptnn04ifc 19115* The function value of a mapping from the nonnegative integers with four distinct cases for the third case. (Contributed by AV, 10-Nov-2019.)

Theoremfvmptnn04ifd 19116* The function value of a mapping from the nonnegative integers with four distinct cases for the forth case. (Contributed by AV, 10-Nov-2019.)

Theoremchfacfisf 19117* The "characteristic factor function" is a function from the nonnegative integers to polynomial matrices. (Contributed by AV, 8-Nov-2019.)
Mat               Poly1       Mat                             matToPolyMat

Theoremchfacfisfcpmat 19118* The "characteristic factor function" is a function from the nonnegative integers to constant polynomial matrices. (Contributed by AV, 19-Nov-2019.)
Mat               Poly1       Mat                             matToPolyMat               ConstPolyMat

Theoremchfacffsupp 19119* The "characteristic factor function" is finitely supported. (Contributed by AV, 20-Nov-2019.) (Proof shortened by AV, 23-Dec-2019.)
Mat               Poly1       Mat                             matToPolyMat               finSupp

Theoremchfacfscmulcl 19120* Closure of a scaled value of the "characteristic factor function". (Contributed by AV, 9-Nov-2019.)
Mat               Poly1       Mat                             matToPolyMat               var1              .gmulGrp

Theoremchfacfscmul0 19121* A scaled value of the "characteristic factor function" is zero almost always. (Contributed by AV, 9-Nov-2019.)
Mat               Poly1       Mat                             matToPolyMat               var1              .gmulGrp

Theoremchfacfscmulfsupp 19122* A mapping of scaled values of the "characteristic factor function" is finitely supported. (Contributed by AV, 8-Nov-2019.)
Mat               Poly1       Mat                             matToPolyMat               var1              .gmulGrp       finSupp

Theoremchfacfscmulgsum 19123* Breaking up a sum of values of the "characteristic factor function" scaled by a polynomial. (Contributed by AV, 9-Nov-2019.)
Mat               Poly1       Mat                             matToPolyMat               var1              .gmulGrp              g g

Theoremchfacfpmmulcl 19124* Closure of the value of the "characteristic factor function" multiplied with a constant polynomial matrix. (Contributed by AV, 23-Nov-2019.)
Mat               Poly1       Mat                             matToPolyMat               .gmulGrp

Theoremchfacfpmmul0 19125* The value of the "characteristic factor function" multiplied with a constant polynomial matrix is zero almost always. (Contributed by AV, 23-Nov-2019.)
Mat               Poly1       Mat                             matToPolyMat               .gmulGrp

Theoremchfacfpmmulfsupp 19126* A mapping of values of the "characteristic factor function" multiplied with a constant polynomial matrix is finitely supported. (Contributed by AV, 23-Nov-2019.)
Mat               Poly1       Mat                             matToPolyMat               .gmulGrp       finSupp

Theoremchfacfpmmulgsum 19127* Breaking up a sum of values of the "characteristic factor function" multiplied with a constant polynomial matrix. (Contributed by AV, 23-Nov-2019.)
Mat               Poly1       Mat                             matToPolyMat               .gmulGrp              g g

Theoremchfacfpmmulgsum2 19128* Breaking up a sum of values of the "characteristic factor function" multiplied with a constant polynomial matrix. (Contributed by AV, 23-Nov-2019.)
Mat               Poly1       Mat                             matToPolyMat               .gmulGrp              g g

Theoremcayhamlem1 19129* Lemma 1 for cayleyhamilton 19153. (Contributed by AV, 11-Nov-2019.)
Mat               Poly1       Mat                             matToPolyMat               .gmulGrp       g

11.5.3  The Cayley-Hamilton theorem

In this section, a direct algebraic proof for the Cayley-Hamilton theorem is provided, according to Wikipedia ("Cayley-Hamilton theorem", 09-Nov-2019, https://en.wikipedia.org/wiki/Cayley%E2%80%93Hamilton_theorem, section "A direct algebraic proof" (this approach is also used for proving Lemma 1.9 in [Hefferon] p. 427):

"This proof uses just the kind of objects needed to formulate the Cayley-Hamilton theorem: matrices with polynomials as entries. The matrix (t * In - A) whose determinant is the characteristic polynomial of A is such a matrix, and since polynomials [over a commutative ring] form a commutative ring, it has an adjugate

(1) B = adj(t * In - A) .

Then, according to the right-hand fundamental relation of the adjugate, one has

(2) ( t * In - A ) x B = det(t * In - A) x In = p(t) * In .

Since B is also a matrix with polynomials in t as entries, one can, for each i, collect the coefficients of t^i in each entry to form a matrix Bi of numbers, such that one has

(3) B = sumi = 0 to (n-1) t^i Bi .

(The way the entries of B are defined makes clear that no powers higher than t^(n-1) occur). While this looks like a polynomial with matrices as coefficients, we shall not consider such a notion; it is just a way to write a matrix with polynomial entries as a linear combination of n constant matrices, and the coefficient t^i has been written to the left of the matrix to stress this point of view.

Now, one can expand the matrix product in our equation by bilinearity

(4) p(t) * In = ( t * In - A ) x B
= ( t * In - A ) x sumi = 0 to (n-1) t^i * Bi
= sumi = 0 to (n-1) t * In x t^i Bi - sumi = 0 to (n-1) A * t^i Bi
= sumi = 0 to (n-1) t^(i+1) * Bi - sumi = 0 to (n-1) t^i * A x Bi
= t^n Bn-1 + sumi = 1 to (n-1) t^i * ( Bi-1 - A x Bi ) - A x B0 .

Writing

(5) p(t) In = t^n * In + t^(n-1) * c(n-1) x In + ... + t * c1 In + c0 In ,

one obtains an equality of two matrices with polynomial entries, written as linear combinations of constant matrices with powers of t as coefficients. Such an equality can hold only if in any matrix position the entry that is multiplied by a given power t^i is the same on both sides; it follows that the constant matrices with coefficient t^i in both expressions must be equal. Writing these equations then for i from n down to 0, one finds

(6) Bn-1 = In , Bi-1 - A x Bi = ci * In for 1 <= i <= n-1 , - A x B0 = c0 * In .

Finally, multiply the equation of the coefficients of t^i from the left by A^i, and sum up:

(7) A^n Bn-1 + sumi = 1 to (n-1) ( A^i x Bi-1 - A^(i+1) x Bi ) - A x B0 = A^n + cn-1 * A^(n-1) + ... + c1 * A + c0 * In .

The left-hand sides form a telescoping sum and cancel completely; the right-hand sides add up to p(A):

(8) 0 = p(A) .

This completes the proof."

To formalize this approach, the steps mentioned in Wikipedia must be elaborated in more detail.

The first step is to formalize the preliminaries and the objective of the theorem. In Wikipedia, the Cayley-Hamilton theorem is stated as follows: "... the Cayley-Hamilton theorem ... states that every square matrix over a commutative ring ... satisfies its own characteristic equation." Or in more detail: "If A is a given n x n matrix and In is the n x n identity matrix, then the characteristic polynomial of A is defined as p(t) = det(t * In - A), where det is the determinant operation and t is a variable for a scalar element of the base ring. Since the entries of the matrix (t * In - A) are (linear or constant) polynomials in t, the determinant is also an n-th order monic polynomial in t. The Cayley-Hamilton theorem states that if one defines an analogous matrix equation, p(A), consisting of the replacement of the scalar eigenvalues t with the matrix A, then this polynomial in the matrix A results in the zero matrix,

p(A) = 0.

The powers of A, obtained by substitution from powers of t, are defined by repeated matrix multiplication; the constant term of p(t) gives a multiple of the power A^0, which is defined as the identity matrix. The theorem allows A^n to be expressed as a linear combination of the lower matrix powers of A. When the ring is a field, the Cayley-Hamilton theorem is equivalent to the statement that the minimal polynomial of a square matrix divides its characteristic polynomial."

Actually, the definition of the characteristic polynomial of a square matrix requires some attention. According to df-chpmat 19090, the characteristic polynomial of an x matrix over a ring is defined as

CharPlyMat

where maDet is the function mapping an x matrix over the polynomial ring over the ring to its determinant, var1 is the variable of the polynomials over , is the x identity matrix as matrix over the polynomial ring over the ring (not the x identity matrix of the matrices over the ring !) and matToPolyMat is the matrix over a ring transformed into a constant matrix over the polynomial ring over the ring . Thus is the multiplication of a polynomial matrix with a "scalar", i.e. a polynomial (see chpmatval 19094).

By this definition, it is assured that , the matrix whose determinat is the characteristic polynomial of the matrix , is actually a matrix over the polynomial ring over the ring , as stated in Wikipedia ("matrix with polynomials as entries"). This matrix is called the characteristic matrix of matrix (see Wikipedia "Polynomial matrix", 16-Nov-2019, https://en.wikipedia.org/wiki/Polynomial_matrix). Following the notation in Wikipedia, we denote the characteristic polynomial of the matrix , formally defined by CharPlyMat as "p(M)" in the comments. The characteristric matrix will be denoted by "c(M)", so that "p(M) = det( c(M) )".

After the preliminaries are clarified, the objective of the Cayley-Hamilton theorem must be considered. As described in Wikipedia, the matrix must be "inserted" into its characteristic polynomial in an appropriate way. Since every polynomial can be represented as infinite, but finitely supported sum of monomials scaled by the corresponding coefficients (see ply1coe 18103), also the characteristic polynomial can be written in this way:

p(M) = sumi ( pi * M^i )

Here, * is the scalar multiplication in the algebra of the polynomials over the ring , and the coefficients are elements of the ring .

By this, we can "define" the insertion of the matrix M into its characteristic polynomial by "p(M) = sum( pi * M^i)", see also cayleyhamilton1 19155. Here, * is the scalar multiplication in the algebra of the matrices over the ring .

To prove the Cayley-Hamilton theorem, we have to show that "p(M) = 0", where 0 is the zero matrix.

In this section, the following class variables and informal identifiers (acronyms in the form "A(B)" or "AB") are used:

class variable/ informal identifier definiens explanation
An arbitrary finite set, used as dimension for matrices
An arbitrary (commutative) ring:
B(R) Base set of (the ring)
Mat Algebra of x matrices over (the ring)
Base set of the algebra of x matrices, i .e. the set of all x matrices
An arbitrary x matrix
Poly1 The algebra of polynomials over (the ring)
B(P) Base set of the algebra of polynomials, i .e. the set of all polynomials
, XR var1 The variable of polynomials over (the ring)
, XA var1 The variable of polynomials over matrices of the algebra
.gmulGrp The group exponentiation for polynomials over (the ring)
^ Arbitrary group exponentiation
algSc The injection of scalars, i.e. elements of (the ring) into the base set of the algebra of polynomials over
, S(p) algSc The element of (the ring) represented as polynomial over
Mat Algebra of x matrices over (the polynomial ring) over the ring
B(Y) Base set of the algebra of polynomial x matrices, i .e. the set of all polynomial x matrices
Poly1 Algebra of polynomials over the ring of x matrices over the ring
B(Q) Base set of the algebra of polynomials over the ring of x matrices over the ring , i .e. the set of all polynomials having x matrices as coefficients
, + The addition of polynomial matrices
, - The subtraction of polynomial matrices
, *Y The multiplication of a polynomial matrix with a scalar ( i. e. a polynomial)
*A The multiplication of a matrix with a scalar ( i. e. an element of the underlying ring)
*Q The multiplication of a polynomial over matrices with a scalar ( i. e. a matrix)
, xY The multiplication of polynomial matrices
xA The multiplication of matrices
xQ The multiplication of polynomials over matrices
, 1Y The identity matrix in the algebra of polynomial matrices over
1A The identity matrix in the algebra of matrices over
, 0Y The zero matrix in the algebra of matrices consisting of polynomials
matToPolyMat The transformation of an x matrix over into a polynomial x matrix over
T1(M) The matrix M transformed into a polynomial x matrix over
U(M) The (constant) polynomial x matrix M transformed into a matrix over the ring . Inverse function of : (see m2cpminvid2 19018 )
T2(M) pMatToMatPoly The polynomial x matrix M transformed into a polynomial over the x matrices over
, c(M) The characteristic matrix of a matrix , i.e. the matrix whose determinant is the characteristic polynomial of the matrix
CharPlyMat The function mapping a matrix over (a ring) to its characteristic polynomial
, p(M) The characteristic polynomial of a matrix over (a ring)
The scalar matrix (diagonal matrix) with the characteristic polynomial of a matrix as diagional elements
maAdju The function mapping a matrix consisting of polynomials to its adjugate ("matrix of cofactors")

Using this notation, we have:
1. "c(M) e. B(Y)", or , see chmatcl 19091
2. "p(M) e. B(P)", or , see chpmatply1 19095
3. "T(M) e. B(Y)", or , see mat2pmatbas 18989

Following the proof shown in Wikipedia, the following steps are performed:
1. Write , the adjugate of the characteristic matrix, as a finite sum of scaled monomials, see pmatcollpw3fi1 19051:
adj(cm(M)) = sumi=0 to s ( XR ^i *Y T1(b(i)) )
where b(i) are matrices over the ring , so T1(b(i)) are constant polynomial matrices.
This step corresponds to (3) in Wikipedia. In contrast to Wikipedia, we write as finite sum of not exactly determined number of summands, which may be greater than needed (including summands of value 0). This will be sufficient to provide a representation of as infinite, but finitely supported sum, see step 3.
2. Write , the product of the characteristic matrix and its adjugate as finite sum of scaled monomials, see cpmadugsumfi 19140. This representation is obtained by replacing by the representation resulting from step 1. and performing calculation rules available for the associative algebra of matrices over polynomials over a commutative ring:
cm(M) *Y adj(cm(M)) = sumi=0 to s ( XR ^i *Y ( T1(b(i-1)) - T1(M) xY T1(b(i)) ) ) + XR ^(s+1) *Y ( T1(b(s)) - T1(M) xY T1(b(0))
where b(i) are matrices over , so T1(b(i)) are constant polynomial matrices:
= cm(M) *Y sumi=0 to s ( XR ^i *Y T1(b(i)) ) [see pmatcollpw3fi1 19051 (step 1.)]
= ( ( XA *Y 1Y ) - T1(M) ) *Y sumi=0 to s ( XR ^i *Y T1(b(i)) ) [def. of cm(M)]
= ( XA *Y 1Y ) *Y sumi=0 to s ( XR ^i *Y T1(b(i)) ) - T1(M) *Y sumi=0 to s ( XR ^i *Y T1(b(i)) ) [see rngsubdir 17027]
= sumi=0 to s ( XR ^i *Y ( T1(b(i-1)) - T1(M) xY T1(b(i)) ) ) + XR ^(s+1) *Y ( T1(b(s)) - T1(M) xY T1(b(0)) [see cpmadugsumlemF 19139]
This step corresponds partially to (4) in Wikipedia.
3. Write as infinite, but finitely supported sum of scaled monomials, see cpmadugsum 19141:
cm(M) * adj(cm(M)) = sumi ( XR ^i *Y G(i) )
This representation is obtained by defining a function G for the coefficients, which we call "characteristic factor function", see chfacfisf 19117, which covers the special terms and the padding with 0. G(i) is a constant polynomial matrix (see chfacfisfcpmat 19118). This step corresponds partially to (4) in Wikipedia, with summands of value 0 added.
4. Write , the scalar matrix (diagonal matrix) with the characteristic polynomial of a matrix as diagional elements, as infinite, but finitely supported sum of scaled monomials. See cpmidgsum 19131:
p(m) *Y IY = sumi ( XR ^i *Y ( S(pi) *Y IY ) )
The proof of cpmidgsum 19131 is making use of pmatcollpwscmat 19054, because is a scalar/diagonal polynomial matrix with the characteristic polynomial "p(M)" as diagonal entries (since pi is an element of the ring , S(pi) is a (constant) polynomial). This corresponds to (5) in Wikipedia, with summands of value 0 added.
5. Transform the sum representation of from step 3. into polynomials over matrices:
T2(cm(M) * adj(cm(M))) = sumi ( U(G(i)) *Q XA ^i ) [see cpmadumatpoly 19146]
where U(G(i)) is a matrix over the ring .
6. Transform the sum representation of from step 4. into polynomials over matrices:
T2(p(m) *Y IY) = sumi ( pi *A IA ) *Q XA ^i ) [see cpmidpmat 19136]
7. Equate the sum representations resulting from steps 5. and 6. by using cpmadurid 19130 to obtain the equation
sumi ( U(G(i)) *Q XA ^i ) = sumi ( pi *A IA ) *Q XA ^i ):
sumi ( U(G(i)) *Q XA ^i )
= T2(cm(M) * adj(cm(M))) [see step 5.]
= T2(p(m) *Y IY) [see cpmadurid 19130]
= sumi ( pi *A IA ) *Q XA ^i ) [see step 6.]
Note that this step is contained in the proof of chcoeffeq 19149, see step 9. This step corresponds to the conclusion from (4) and (5) in Wikipedia, with summands of value 0 added.
8. Compare the sum representations of step 7. to obtain the equations U(G(i)) = pi *A IA , see chcoeffeqlem 19148. This corresponds to (6) in Wikipedia. Since the coefficients of the transformed representations and the original representations are identical, the equations of the coefficients are also valid for the original representations of steps 3. and 4.
9. Multiply the equations of the coefficients from step 8. from the left by M^i, and sum up, see chcoeffeq 19149:
sumi ( M^i xA U(G(i)) ) = sumi ( M^i xA ( pi *A IA) )
This corresponds to (7) in Wikipedia.
10. Transform the right hand side of the equation in step 9. into an appropriate form, see cayhamlem3 19150:
sumi ( pi *A M^i )
= sumi ( M^i xA ( pi *A IA) ) [see cayhamlem2 19147]
= sumi ( M^i xA U(G(i)) ) [see chcoeffeq 19149]
11. Apply the theorem for telescoping sums, see telgsumfz 16805, to the sum sumi ( T1(M)^i xY G(i) ), which results in an equation to 0:
sumi ( T1(M)^i xY G(i) ) = 0Y, see cayhamlem1 19129:
sumi ( T1(M)^i xY G(i) )
= sumi=1 to s ( T1(M)^i xY T1(b(i-1)) - T1(M)^(i+1) xY T1(b(i)) )
+ ( T1(M)^(s+1) xY T1(b(s)) - T1(M) xY T1(b(0)) ) [see chfacfpmmulgsum2 19128]
= ( T1(M) xY T1(b(0)) - T1(M)^(s+1) xY T1(b(s)) ) + ( T1 M)^(s+1) xY T1(b(s)) - T1(M) xY T1(b(0)) ) [see telgsumfz 16805]
= 0Y [see grpnpncan0 15930] This step corresponds partially to (8) in Wikipedia.
12. Since is a ring homomorphism (see mat2pmatrhm 18997), the left hand side of the equation in step 10. can be transformed into a representation appropriate to apply the result of step 11., see cayhamlem4 19151:
sumi ( pi *A M^i )
= sumi ( M^i xA U(G(i)) ) [see cayhamlem3 19150 (step 10.)]
= U(T1(sumi ( M^i xA U(G(i)) ))) [see m2cpminvid 19016]
= U(sumi T1( M^i xA U(G(i)) )) [see gsummptmhm 16749]
= U(sumi ( T1(M^i) xY T1(U(G(i))) )) [see rhmmul 17155]
= U(sumi ( T1(M)^i xY T1(U(G(i))) )) [see mhmmulg 15969]
= U(sumi ( T1(M)^i xY G(i) )) [see m2cpminvid2 19018 ]
13. Finally, combine the results of steps 11. and 12., and use the fact that (and therefore also its inverse ) is an injective ring homomorphism (see mat2pmatf1 18992 and mat2pmatrhm 18997) to transform the equality resulting from steps 11. and 12. into the desired equation sumi ( pi *A M^i ) = 0A , see cayleyhamilton 19153 resp. cayleyhamilton0 19152:
sumi ( pi *A M^i )
= U(sumi ( T1(M)^i xY G(i) )) [see cayhamlem4 19151 (step 12.)]
= U(0Y ) [see cayhamlem1 19129 (step 11.)]
= 0A [see m2cpminv0 19024]
The transformations in steps 5., 6., 10., 12. and 13. are not mentioned in the proof provided in Wikipedia, since it makes no distinction between a matrix over a ring and its representation as matrix over the polynomial ring over the ring in general!

Theoremcpmadurid 19130 The right-hand fundamental relation of the adjugate (see madurid 18908) applied to the characteristic matrix of a matrix. (Contributed by AV, 25-Oct-2019.)
Mat               CharPlyMat        Poly1       Mat        var1       matToPolyMat                                    maAdju

Theoremcpmidgsum 19131* Representation of the identity matrix multiplied with the characteristic polynomial of a matrix as group sum. (Contributed by AV, 7-Nov-2019.)
Mat               Poly1       Mat        var1       .gmulGrp                     algSc       CharPlyMat                      g coe1

Theoremcpmidgsumm2pm 19132* Representation of the identity matrix multiplied with the characteristic polynomial of a matrix as group sum with a matrix to polynomial matrix transformation. (Contributed by AV, 13-Nov-2019.)
Mat               Poly1       Mat        var1       .gmulGrp                     algSc       CharPlyMat                                    matToPolyMat        g coe1

Theoremcpmidpmatlem1 19133* Lemma 1 for cpmidpmat 19136. (Contributed by AV, 13-Nov-2019.)
Mat               Poly1       Mat        var1       .gmulGrp                     algSc       CharPlyMat                                    matToPolyMat        coe1        coe1

Theoremcpmidpmatlem2 19134* Lemma 2 for cpmidpmat 19136. (Contributed by AV, 14-Nov-2019.) (Proof shortened by AV, 7-Dec-2019.)
Mat               Poly1       Mat        var1       .gmulGrp                     algSc       CharPlyMat                                    matToPolyMat        coe1

Theoremcpmidpmatlem3 19135* Lemma 3 for cpmidpmat 19136. (Contributed by AV, 14-Nov-2019.) (Proof shortened by AV, 7-Dec-2019.)
Mat               Poly1       Mat        var1       .gmulGrp                     algSc       CharPlyMat                                    matToPolyMat        coe1        finSupp

Theoremcpmidpmat 19136* Representation of the identity matrix multiplied with the characteristic polynomial of a matrix as polynomial over the ring of matrices. (Contributed by AV, 14-Nov-2019.) (Revised by AV, 7-Dec-2019.)
Mat               Poly1       Mat        var1       .gmulGrp                     algSc       CharPlyMat                                    matToPolyMat               Poly1       var1              .gmulGrp       pMatToMatPoly        g coe1

Mat               Poly1       Mat        matToPolyMat        var1       .gmulGrp                            g g

Mat               Poly1       Mat        matToPolyMat        var1       .gmulGrp                            g g

Mat               Poly1       Mat        matToPolyMat        var1       .gmulGrp                                          g g g

Theoremcpmadugsumfi 19140* The product of the characteristic matrix of a given matrix and its adjunct represented as finite sum. (Contributed by AV, 7-Nov-2019.) (Proof shortened by AV, 29-Nov-2019.)
Mat               Poly1       Mat        matToPolyMat        var1       .gmulGrp                                                 maAdju        g

Theoremcpmadugsum 19141* The product of the characteristic matrix of a given matrix and its adjunct represented as an infinite sum. (Contributed by AV, 10-Nov-2019.)
Mat               Poly1       Mat        matToPolyMat        var1       .gmulGrp                                                 maAdju                      g

Theoremcpmidgsum2 19142* Representation of the identity matrix multiplied with the characteristic polynomial of a matrix as another group sum. (Contributed by AV, 10-Nov-2019.)
Mat               Poly1       Mat        matToPolyMat        var1       .gmulGrp                                                 maAdju                      CharPlyMat                      g

Theoremcpmidg2sum 19143* Equality of two sums representing the identity matrix multiplied with the characteristic polynomial of a matrix. (Contributed by AV, 11-Nov-2019.)
Mat               Poly1       Mat        matToPolyMat        var1       .gmulGrp                                                 maAdju                      CharPlyMat               algSc       g coe1 g

Theoremcpmadumatpolylem1 19144* Lemma 1 for cpmadumatpoly 19146. (Contributed by AV, 20-Nov-2019.) (Revised by AV, 15-Dec-2019.)
Mat               Poly1       Mat        matToPolyMat                                    ConstPolyMat                      var1              maAdju               Poly1       var1              .gmulGrp       cPolyMatToMat

Theoremcpmadumatpolylem2 19145* Lemma 2 for cpmadumatpoly 19146. (Contributed by AV, 20-Nov-2019.) (Revised by AV, 15-Dec-2019.)
Mat               Poly1       Mat        matToPolyMat                                    ConstPolyMat                      var1              maAdju               Poly1       var1              .gmulGrp       cPolyMatToMat        finSupp

Theoremcpmadumatpoly 19146* The product of the characteristic matrix of a given matrix and its adjunct represented as a polynomial over matrices. (Contributed by AV, 20-Nov-2019.) (Revised by AV, 7-Dec-2019.)
Mat               Poly1       Mat        matToPolyMat                                    ConstPolyMat                      var1              maAdju               Poly1       var1              .gmulGrp       cPolyMatToMat        pMatToMatPoly        g

Theoremcayhamlem2 19147 Lemma for cayhamlem3 19150. (Contributed by AV, 24-Nov-2019.)
Mat                             .gmulGrp

Theoremchcoeffeqlem 19148* Lemma for chcoeffeq 19149. (Contributed by AV, 21-Nov-2019.) (Proof shortened by AV, 7-Dec-2019.) (Revised by AV, 15-Dec-2019.)
Mat               Poly1       Mat                             matToPolyMat        CharPlyMat                                           cPolyMatToMat        Poly1 g Poly1.gmulGrpPoly1var1 Poly1 g coe1 Poly1.gmulGrpPoly1var1 coe1

Theoremchcoeffeq 19149* The coefficients of the characteristic polynomial multiplied with the identity matrix represented by (transformed) ring elements obtained from the adjunct of the characteristic matrix. (Contributed by AV, 21-Nov-2019.) (Proof shortened by AV, 8-Dec-2019.) (Revised by AV, 15-Dec-2019.)
Mat               Poly1       Mat                             matToPolyMat        CharPlyMat                                           cPolyMatToMat        coe1

Theoremcayhamlem3 19150* Lemma for cayhamlem4 19151. (Contributed by AV, 24-Nov-2019.) (Revised by AV, 15-Dec-2019.)
Mat               Poly1       Mat                             matToPolyMat        CharPlyMat                                           cPolyMatToMat        .gmulGrp              g coe1 g

Theoremcayhamlem4 19151* Lemma for cayleyhamilton 19153. (Contributed by AV, 25-Nov-2019.) (Revised by AV, 15-Dec-2019.)
Mat               Poly1       Mat                             matToPolyMat        CharPlyMat                                           cPolyMatToMat        .gmulGrp       .gmulGrp       g coe1 g

Theoremcayleyhamilton0 19152* The Cayley-Hamilton theorem: A matrix over a commutative ring "satisfies its own characteristic equation". This version of cayleyhamilton 19153 provides definitions not used in the theorem itself, but in its proof to make it clearer, more readable and shorter compared with a proof without them (see cayleyhamiltonALT 19154)! (Contributed by AV, 25-Nov-2019.) (Revised by AV, 15-Dec-2019.)
Mat                                    .gmulGrp       CharPlyMat        coe1       Poly1       Mat                                    .gmulGrp       matToPolyMat               cPolyMatToMat        g

Theoremcayleyhamilton 19153* The Cayley-Hamilton theorem: A matrix over a commutative ring "satisfies its own characteristic equation", see theorem 7.8 in [Roman] p. 170 (without proof!), or theorem 3.1 in [Lang] p. 561. In other words, a matrix over a commutative ring "inserted" into its characteristic polynomial results in zero. This is Metamath 100 proof #49. (Contributed by Alexander van der Vekens, 25-Nov-2019.)
Mat                      CharPlyMat        coe1              .gmulGrp       g

TheoremcayleyhamiltonALT 19154* Alternate proof of cayleyhamilton 19153, the Cayley-Hamilton theorem. This proof does not use cayleyhamilton0 19152 directly, but has the same structure as the proof of cayleyhamilton0 19152. In contrast to the proof of cayleyhamilton0 19152, only the definitions required to formulate the theorem itself are used, causing the definitions used in the lemmata being expanded, which makes the proof longer and more difficult to read. (Contributed by AV, 25-Nov-2019.) (New usage is discouraged.) (Proof modification is discouraged.)
Mat                      CharPlyMat        coe1              .gmulGrp       g

Theoremcayleyhamilton1 19155* The Cayley-Hamilton theorem: A matrix over a commutative ring "satisfies its own characteristic equation", or, in other words, a matrix over a commutative ring "inserted" into its characteristic polynomial results in zero. In this variant of cayleyhamilton 19153, the meaning of "inserted" is made more transparent: If the characteristic polynomial is a polynomial with coefficients , then a matrix over a commutative ring "inserted" into its characteristic polynomial is the sum of these coefficients multiplied with the corresponding power of the matrix. (Contributed by AV, 25-Nov-2019.)
Mat                      CharPlyMat        coe1              .gmulGrp              var1       Poly1              .gmulGrp              finSupp g g

PART 12  BASIC TOPOLOGY

12.1  Topology

12.1.1  Topological spaces

Syntaxctop 19156 Extend class notation with the class of all topologies.

Syntaxctopon 19157 The class function of all topologies over a base set.
TopOn

SyntaxctpsOLD 19158 Extend class notation with the class of all topological spaces. (New usage is discouraged.)

Syntaxctps 19159 Extend class notation with the class of all topological spaces.

Syntaxctb 19160 Extend class notation with the class of all topological bases.

Definitiondf-top 19161* Define the (proper) class of all topologies. See istop2g 19167 for an alternate way to express finite intersection and istps5OLD 19187 for a standard definition as an ordered pair of a set and a topology on it.

The final form of the definition is due to Bourbaki (Def. 1 of [BourbakiTop1] p. I.1), while the idea of defining a topology in terms of its open sets is due to Aleksandrov. For the convoluted history of the definitions of these notions, see

Gregory H. Moore, The emergence of open sets, closed sets, and limit points in analysis and topology, Historia Mathematica 35 (2008) 220--241.

(Contributed by NM, 3-Mar-2006.) (Revised by BJ, 20-Oct-2018.)

Definitiondf-topspOLD 19162* Define the class of all topological spaces, each of which is an ordered pair the second of which is a topology on the first. See istps5OLD 19187 for a standard way to express a topological space. (Contributed by NM, 8-Mar-2006.) (New usage is discouraged.)

Definitiondf-bases 19163* Define the class of all topological bases. Equivalent to definition of basis in [Munkres] p. 78 (see isbasis2g 19211). Note that "bases" is the plural of "basis." (Contributed by NM, 17-Jul-2006.)

Definitiondf-topon 19164* Define the set of topologies with a given base set. (Contributed by Stefan O'Rear, 31-Jan-2015.)
TopOn

Definitiondf-topsp 19165 Define the class of all topological spaces (structures). (Contributed by Stefan O'Rear, 13-Aug-2015.)
TopOn

Theoremistopg 19166* Express the predicate " is a topology." Note: In the literature, a topology is often represented by a script letter T, which resembles the letter J. This confusion may have led to J being used by some authors - e.g. K. D. Joshi, Introduction to General Topology (1983), p. 114 - and it is convenient for us since we later use to represent linear transformations (operators). (Contributed by Stefan Allan, 3-Mar-2006.) (Revised by Mario Carneiro, 11-Nov-2013.)

Theoremistop2g 19167* Express the predicate " is a topology," using "the intersection of the elements of any finite subcollection" instead of the intersection of any two elements. (Contributed by NM, 19-Jul-2006.)

Theoremuniopn 19168 The union of a subset of a topology is an open set. (Contributed by Stefan Allan, 27-Feb-2006.)

Theoremiunopn 19169* The indexed union of a subset of a topology is an open set. (Contributed by NM, 5-Oct-2006.)

Theoreminopn 19170 The intersection of two open sets of a topology is also an open set. (Contributed by NM, 17-Jul-2006.)

Theoremfitop 19171 A topology is closed under finite intersections. (Contributed by Jeff Hankins, 7-Oct-2009.)

Theoremfiinopn 19172 The intersection of a nonempty finite family of open sets is open. (Contributed by FL, 20-Apr-2012.)

Theoremiinopn 19173* The intersection of a nonempty finite family of open sets is open. (Contributed by Mario Carneiro, 14-Sep-2014.)

Theoremunopn 19174 The union of two open sets is open. (Contributed by Jeff Madsen, 2-Sep-2009.)

Theorem0opn 19175 The empty set is an open subset of a topology. (Contributed by Stefan Allan, 27-Feb-2006.)

Theorem0ntop 19176 The empty set is not a topology. (Contributed by FL, 1-Jun-2008.)

Theoremtopopn 19177 The underlying set of a topology is an open set. (Contributed by NM, 17-Jul-2006.)

Theoremeltopss 19178 A member of a topology is a subset of its underlying set. (Contributed by NM, 12-Sep-2006.)

Theoremriinopn 19179* A finite indexed relative intersection of open sets is open. (Contributed by Mario Carneiro, 22-Aug-2015.)

Theoremrintopn 19180 A finite relative intersection of open sets is open. (Contributed by Mario Carneiro, 22-Aug-2015.)

TheoremeltopspOLD 19181 Construct a topological space from a topology and vice-versa. We say that is a topology on . (This could be proved more efficiently from istpsOLD 19183, but the proof here does not require the Axiom of Regularity.) (Contributed by NM, 8-Mar-2006.) (Proof modification is discouraged.) (New usage is discouraged.)

TheoremtpsexOLD 19182 Existence implied by membership in a topological space. (Contributed by NM, 18-Jul-2006.) (Revised by Mario Carneiro, 30-Apr-2015.) (Proof modification is discouraged.) (New usage is discouraged.)

TheoremistpsOLD 19183 Express the predicate "is a topological space." (Contributed by NM, 18-Jul-2006.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremistps2OLD 19184 Express the predicate "is a topological space." (Contributed by NM, 18-Jul-2006.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremistps3OLD 19185* A standard textbook definition of a topological space. (Contributed by NM, 18-Jul-2006.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremistps4OLD 19186* A standard textbook definition of a topological space. (Contributed by NM, 19-Jul-2006.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremistps5OLD 19187* A standard textbook definition of a topological space : a topology on is a collection of subsets of such that and are in and that is closed under union and finite intersection. Definition of topological space in [Munkres] p. 76. (Contributed by NM, 19-Jul-2006.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremistopon 19188 Property of being a topology with a given base set. (Contributed by Stefan O'Rear, 31-Jan-2015.) (Revised by Mario Carneiro, 13-Aug-2015.)
TopOn

Theoremtopontop 19189 A topology on a given base set is a topology. (Contributed by Mario Carneiro, 13-Aug-2015.)
TopOn

Theoremtoponuni 19190 The base set of a topology on a given base set. (Contributed by Mario Carneiro, 13-Aug-2015.)
TopOn

Theoremtoponmax 19191 The base set of a topology is an open set. (Contributed by Mario Carneiro, 13-Aug-2015.)
TopOn

Theoremtoponss 19192 A member of a topology is a subset of its underlying set. (Contributed by Mario Carneiro, 21-Aug-2015.)
TopOn

Theoremtoponcom 19193 If is a topology on the base set of topology , then is a topology on the base of . (Contributed by Mario Carneiro, 22-Aug-2015.)
TopOn TopOn

Theoremtopontopi 19194 A topology on a given base set is a topology. (Contributed by Mario Carneiro, 13-Aug-2015.)
TopOn

Theoremtoponunii 19195 The base set of a topology on a given base set. (Contributed by Mario Carneiro, 13-Aug-2015.)
TopOn

Theoremtoptopon 19196 Alternative definition of in terms of TopOn. (Contributed by Mario Carneiro, 13-Aug-2015.)
TopOn

Theoremtopgele 19197 The topologies over the same set have the greatest element (the discrete topology) and the least element (the indiscrete topology). (Contributed by FL, 18-Apr-2010.) (Revised by Mario Carneiro, 16-Sep-2015.)
TopOn

Theoremtopsn 19198 The only topology on a singleton is the discrete topology (which is also the indiscrete topology by pwsn 4234). (Contributed by FL, 5-Jan-2009.) (Revised by Mario Carneiro, 16-Sep-2015.)
TopOn

Theoremistps 19199 Express the predicate "is a topological space." (Contributed by Mario Carneiro, 13-Aug-2015.)
TopOn

Theoremistps2 19200 Express the predicate "is a topological space." (Contributed by NM, 20-Oct-2012.)

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