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Congruence, Congruent Transformation, Symmetric matrices, Skew-symmetric matrices, Hermitian matrices, Skew-Hermitian matrices



Congruent Transformation. A transformation of the form B = PTAP of a matrix A by a non-singular matrix P, where PT is the transpose of P. B is said to be congruent to A.


Two n-square matrices A and B over some field F (real numbers, complex numbers, etc) are called congruent over F if there exists a nonsingular matrix P over F such that B = PTAP .


Since congruence is a special case of equivalence, congruent matrices have the same rank.


Theorem 1. Matrices A and B are congruent provided A can be reduced to B by a sequence of pairs of elementary operations, each pair consisting of an elementary row operation followed by the same elementary column operation.


Proof. Let P be expressed as a product of elementary column matrices:


                                      P = K1K2 ... Kn


Then PT = KnT ... K2TK1T . Now because of the fact that an elementary row matrix is the transpose of the corresponding elementary column matrix


                                  KnT ... K2TK1T = Hn ... H2H1


where Hi is the elementary row operation corresponding to the elementary column operation Ki.. Thus


                                  PT = Hn ... H2H1


and


                  B = PTAP = Hn ... H2H1A K1K2 ... Kn

  

which is what we wanted to prove.



The concept of congruence has importance in connection with quadratic forms, specifically in regard to the question of what changes in coordinate system might reduce a particular quadratic form to its simplest form. Consider the quadratic form


             ole.gif



                                       ole1.gif ole2.gif   ole3.gif   



                                   = XTAX


where A is the symmetric matrix of the coefficients of the form and X is the column vector of the variables. The elements aij are elements of some field F (real numbers, complex numbers, etc.). Let us consider what effect a change of basis given by X = PY has on the quadratic form. If X = PY then XT = YTPT and


           XTAX = YTPTAPY


Thus the matrix A is transformed into a congruent matrix under this change of basis. Every symmetric matrix is congruent to a diagonal matrix, and hence every quadratic form can be changed to a form of type ∑kixi2 (its simplest canonical form) by a change of basis.






Symmetric matrices.


1] Every symmetric matrix A over a field F of rank r is congruent over F to a diagonal matrix whose first r diagonal elements are non-zero while all other elements are zero.






Real symmetric matrices.


1] A real symmetric matrix of rank r is congruent over the field of real numbers to a canonical matrix


                                      ole4.gif


The integer p is called the index of the matrix and s = p - (r - p) is called the signature.

 

The index of a symmetric or Hermitian matrix is the number of positive elements when it is transformed to a diagonal matrix. The signature is the number of positive terms diminished by the number of negative terms and the total number of nonzero terms is the rank.


 

2] Two n-square real symmetric matrices are congruent over the real field if and only if they have the same rank and the same index or the same rank and the same signature.





Complex symmetric matrices.


1] Every n-square complex symmetric matrix of rank r is congruent over the field of complex numbers to a canonical matrix


                                            ole5.gif




2] Two n-square complex symmetric matrices are congruent over the field of complex numbers if and only if they have the same rank.






Skew-symmetric matrices.


1] Every matrix B = PTAP congruent to a skew-symmetric matrix A is also skew-symmetric.


2] Every n-square skew-symmetric matrix A over field F is congruent over F to a canonical matrix


                        B = diag(D1, D2, ....., Dt, 0, .... .0)


where


    ole6.gif       


The rank of A is r = 2t.


 


3] Two n-square skew-symmetric matrices over a field F are congruent over F if and only if they have the same rank.


4] The set of all matrices of the type


                                B = diag(D1, D2, ....., Dt, 0, .... .0)


described above is a canonical set over congruence for n-square skew-symmetric matrices.





Hermitian matrices.



Hermitely congruent matrices. Two n-square Hermitian matrices A and B are called Hermitely congruent or conjunctive if there exits a nonsingular matrix P such that


                                     ole7.gif



1] Two n-square Hermitian matrices are conjunctive if and only if one can be obtained from the other by a sequence of pairs of elementary operations, each pair consisting of a column operation and the corresponding conjugate row operation. (For the three column operations Kij, Ki(k), Kij(k) the conjugate row operations are ole8.gif .) The proof is similar to that for Theorem 1 above.


2] An Hermitian matrix A of rank r is conjunctive to a canonical matrix


                                     ole9.gif


The integer p is called the index of the matrix and s = p - (r - p) is called the signature.


3] Two n-square Hermitian matrices are conjunctive if and only if they have the same rank and index or the same rank and signature.





Skew-Hermitian matrices.


1] Every matrix ole10.gif conjunctive to a skew-Hermitian matrix A is also skew-Hermitian.


2] Every n-square skew-Hermitian matrix A is conjunctive to a matrix


                                     ole11.gif


in which r is the rank of A and p is the index of -iA.



3] Two n-square skew-Hermitian matrices A and B are conjunctive if and only if they have the same rank while -iA and -iB have the same index.




References.

  Ayres. Matrices (Schaum).



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