Is any non symmetric matrix which is $M^M =MM^$?

Question

We know $n\times n$ symmetric matrix has $n$ real eigenvalues, but I just wondering whether there are some non symmetric matrix which is $M^*M = MM^*$

score 4 · Accepted Answer · answered Jul 28 '13 at 19:41

Yes, there are many. We call these types of matrices normal matrices. Two instances of these types of matrices are symmetric matrices and antisymmetric matrices.

When dealing with matrices with complex entries, the criteria change and we talk about whether a matrix is Hermitian or anti-Hermitian.

score 1 · Answer 2 · answered Jul 28 '13 at 19:45

Such matrices are called normal and are, among other things, characterized by being orthonormally diagonalizable, which means that their eigenvectors can be chosen in a way to form an orthonormal basis of the vector space.

Typical examples are Hermitian (or, in real case, symmetric), skew-Hermitian (or, in real case, skew-symmetric), and unitary (or, in real case, orthogonal) matrices.

Is any non symmetric matrix which is $M^*M =MM^*$?

2 Answers2

Is any non symmetric matrix which is $M^M =MM^$?