Matrix A multiply A transpose

Question

Suppose $A$ is an $n\times n$ matrix, and wondering when $A A^T$ equals $A^T A$? Only when $A$ is symmetric?

regards, Lin

Answer 1

Symmetric matrices are not the only matrices that satisfy this property. There is a definition for the matrix that you describe:

If $A$ is a complex matrix that satisfies $A^{\ast}A=AA^{\ast}$, then we say A is a normal matrix. Note that $A^{\ast}$ represents $A$ adjoint, i.e. the complex conjugate transpose of $A$.

So if $A$ is just a real matrix and $A$ satisfies $A^tA=AA^t$, then $A$ is a normal matrix, as the complex conjugate transpose of a real matrix is just the transpose of that matrix.

For example, a $2\times 2$ counterclockwise rotation matrix by an angle $\theta$ is given by $B=\begin{bmatrix} \cos\theta&-\sin\theta \\ \sin\theta& \cos\theta \end{bmatrix}$. Notice $B$ is not a symmetric matrix, but does satisfy $B^tB=BB^t$.

Answer 2

Suppose $A \in M_{n,n}(\mathbb{R})$ then

$$AA^T = A^TA \iff A \textrm{ is normal}$$

Answer 3

Matrices with that property are called normal matrices.

See Normal Matrix