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1、Linear AlgebraCriterion of Congruent MatricesSince replacing X in f(X)X TAX by XPY,we get Hence the formula(3)described in the preceding section can be obtained if there exists the invertible matrix P such thatHere D is a diagonal matrix.In other words,reducing a quadratic form to its standard form
2、is equivalent to obtain the matrix representation(4)for the real symmetric matrix A,or by the terminology in the following definition,finding a diagonal matrix D which is congruent to A.Congruent MatricesCongruent MatricesDefinition.Let A and B be two square matrices of order n.If there exists an in
3、vertible matrix P such that BPTAP,then the two matrices A and B are called congruent,and it is denoted by AB.where“T”denotes the transpose of matrix.For example,if A and B are orthogonal similar,then there exists orthogonal matrix P such thatClearly,A and B are orthogonally congruent.If A is real sy
4、mmetric matrix,there exists orthogonal matrix Q,such that where is a diagonal matrix.Therefore the real symmetric matrix and a diagonal matrix are congruent.Proposition.Matrix congruence is an equivalent relation.We often use the following methods to determine whether or not two matrices A and B are
5、 congruent.Theorem.The quadratic form f X TAX can be reduced to the quadratic form f Y TAY by means of a nonsingular linear transformation XPY,then BPTAP is congruent to A.Theorem.A real symmetric matrix is congruent to a diagonal matrix,i.e.,if a matrix A is a real symmetric matrix of rank r,then t
6、here is a nonsingular matrix P such that where ai0,i1,2,r.It should be noted that if A is not a symmetric matrix,then PTAP may not be a diagonal matrix.Theorem.Two real symmetric matrices of order n are congruent if and only if they have the same rank and positive(or negative)inertia index.Theorem.I
7、f two real symmetric matrices are similar,then they must be congruent,but their inverses dont hold.Theorem.Two positive definite matrices of the same order must be congruent.Example.Suppose that A and B are real symmetric matrices of order n,prove that if A and B are congruent,then r(A)r(B).Converse
8、ly,if r(A)r(B),determine whether or not A and B are congruent?Solution.Because A and B are congruent,there exists invertible matrix P such that BPTAP.Hence r(B)r(PTAP)r(A).Conversely,if only r(A)r(B),then A and B may not be congruent.This is because they may be not symmetric matrices.For example,but B is not a symmetric matrix,it can not be congruent to A.