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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 = P^{T}AP of a matrix A by
a non-singular matrix P, where P^{T} 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 = P^{T}AP .

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 = K_{1}K_{2} ... K_{n}

Then P^{T} = K_{n}^{T} ... K_{2}^{T}K_{1}^{T} . Now because of the fact that an elementary row matrix is the
transpose of the corresponding elementary column matrix

K_{n}^{T} ... K_{2}^{T}K_{1}^{T} = H_{n} ... H_{2}H_{1}

where H_{i} is the elementary row operation corresponding to the elementary column operation K_{i}..
Thus

P^{T} = H_{n} ... H_{2}H_{1}

and

B = P^{T}AP = H_{n} ... H_{2}H_{1}A K_{1}K_{2} ... K_{n}

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

= X^{T}AX

where A is the symmetric matrix of the coefficients of the form and X is the column vector of the
variables. The elements a_{ij} 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 X^{T} = Y^{T}P^{T} and

X^{T}AX = Y^{T}P^{T}APY

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 ∑k_{i}x_{i}^{2} (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

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

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 = P^{T}AP 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(D_{1}, D_{2}, ....., D_{t}, 0, .... .0)

where

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(D_{1}, D_{2}, ....., D_{t}, 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

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 K_{ij}, K_{i}(k), K_{ij}(k)
the conjugate row operations are
.) The proof is similar to that for Theorem
1 above.

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

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 conjunctive to a skew-Hermitian matrix A is also skew-Hermitian.

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

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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