实对称矩阵的逆的转置矩阵等于它的逆矩阵吗 对称矩阵的逆矩阵和转置矩阵关系
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等于，因为他的逆也是对称矩阵，注意到转置和逆是可交换的，也就是(A^-1)^T=(A^T)^(-1)，因为A是对称的，故(A^-1)^T=A^(-1)得证。
实对称矩阵A的不同特征值对应的特征向量是正交的。实对称矩阵A的特征值都是实数，特征向量都是实向量。n阶实对称矩阵A必可对角化，且相似对角阵上的元素即为矩阵本身特征值。
扩展资料：
将A的所有元素绕着一条从第1行第1列元素出发的右下方45度的射线作镜面反转，即得到A的转置。设A为m×n阶矩阵（即m行n列），第i 行j 列的元素是a(i,j)，即：A=a(i,j)。
定义A的转置为这样一个n×m阶矩阵B，满足B=b(j,i)，即 a(i,j)=b (j,i)（B的第i行第j列元素是A的第j行第i列元素），记A'=B。
如果矩阵A和B互逆，则AB=BA=I。由条件AB=BA以及矩阵乘法的定义可知，矩阵A和B都是方阵。再由条件AB=I以及定理“两个矩阵的乘积的行列式等于这两个矩阵的行列式的乘积”可知，这两个矩阵的行列式都不为0。
参考资料来源：百度百科——实对称矩阵
等于，因为他的逆也是对称矩阵
注意到转置和逆是可交换的，也就是(A^-1)^T=(A^T)^(-1)
因为A是对称的，故(A^-1)^T=A^(-1)得证。
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