3.矩阵的合同 f=xAx==(Cy)'A(Cy)=y(CAC)y=yBy 定理1.若∫=x4x经可逆线性变换x=Cy 变成∫=yB,则 (1)B=CAC,且B为对称阵; (2)r(A)=r(B) Proof.(1)∵A为实对称阵, B′=(CAC)=CAC=CAC=B (2)由C可逆,可知C可逆;又CAC=B, A与B等价,故r(4)=r(B) K心3. 矩阵的合同 f = xAx (Cy) A(Cy) x Cy ====  = = y(CAC) y = yBy 定理1. 变成 则 若 经可逆线性变换 f y By, f x Ax x Cy =  =  = (1) B = CAC,且B为对称阵; (2) r(A) = r(B). Proof. (1)  A为实对称阵, B = (CAC) = CAC = CAC = B. (2) 由C 可逆,可知C可逆; 又 CAC = B,  A与 B等价, 故 r(A) = r(B)