Product of a Matrix with its Transpose is Symmetric

Theorem

Let $A$ : Matrix

Then $A{A}^T$ and $A^{T}A$ is symmetric

Proof

Let $A$ be a $m\times n$ matrix. Then $A{A}^T$ is a $m\times m$ square matrix and $A^{T}A$ is a $n\times n$ square matrix.

By point 4 and 1 of properties of transpose matrix,

$$\begin{array}{rl}(A{A}^T{)}^T& =(A^T{)}^T{A}^T\\ & =A{A}^T\end{array}$$

and

$$\begin{array}{rl}(A^{T}A{)}^T& =A^T(A^T{)}^T\\ & =A^{T}A.\end{array}$$

Thus by defininition of symmetric matrix, $A{A}^T$ and $A^{T}A$ is symmetric.