Are the row space and the column space of a symmetric matrix identical? And what is the relationship to a matrix of eigenvectors and the transpose of the eigenvectors?
Thanks.
[Math] Column and row space of a symmetric matrix
eigenvalues-eigenvectorslinear algebra
Best Answer
Indeed, for a symmetric matrix, the column space is identical to the row-space. This follows fairly immediately from the respective definitions.
The matrix $P$ of eigenvectors of a symmetric matrix $A$ satisfies $$ A = PDP^{-1} \implies AP = PD $$ Where $D$ is the diagonal matrix of (real) eigenvalues. It follows then that $$ (AP)^T = P^T A^T = P^T A = \\ (PD)^T = D^T P^T = D P^T $$ That is, $P^TA = D P^T$, so that $P^T$ is a matrix of left eigenvectors.