Question

If A is symmetric matrix, then B^TAB is ___________.

Answer 1

It is given that, A is symmetric matrix.

$\therefore A^T=A$ .....(1)

Now,

${\left(B^{T}AB\right)}^T$

$=B^T{A}^T{\left(B^T\right)}^T$ [For any matrices X, Y, Z, (XYZ)^T = Z^TY^TX^T]

$=B^{T}AB$ [Using (1)]

Since ${\left(B^{T}AB\right)}^T$$=B^{T}AB$, so the matrix B^TAB is symmetric.

If A is symmetric matrix, then B^TAB is __symmetric__.