Is matrix $A$ contained in matrix $B$?

Question

$B$ = \begin{bmatrix} a_{11}^2 + a_{21}^2 & a_{11}a_{12} + a_{21}a_{22} \\ a_{12}a_{11} + a_{22}a_{21} & a_{12}^2 + a_{22}^2\end{bmatrix}

$A$ = \begin{bmatrix}a_{11} & a_{12} \\ a_{21} & a_{22}\end{bmatrix}

Can anyone see if matrix $A$ is somehow embedded in matrix $B$?

If not, can matrix $B$ be expressed in a component form, or as an outer product?

Answer 1

Yes, actually it holds

$$B = A^T \cdot A.$$