If a matrix is symmetric then, is similar matrix symmetric also?

Question

A question regarding real similar matrices.

I know if $A\sim B$ then $A=P BP^{-1}$.

But if $A^T=A\geq0$, should be said that $B$ is symmetric and $P=P^{-T}$,
or can I have another choice?

Answer 1

No. Try some examples: take a random symmetric $A$ and a random invertible $P$.

Answer 2

For a simple example, check at https://www.wolframalpha.com

$$ \begin{bmatrix} 1 & 1 \\ 1 & 0 \end{bmatrix} \begin{bmatrix} 1 & 0 \\ 0 & 2 \end{bmatrix} \begin{bmatrix} 1 & 1 \\ 1 & 0 \end{bmatrix}^{-1} $$

{{1,1},{1,0}}*{{1,0},{0,2}}*inverse{{1,1},{1,0}}

(insert directly above text into wolphram textbox)

However
if your matrix $P$ were chosen as orthogonal i.e. $PP^T=I$
then $A=PBP^T$ and $A^T=PB^TP^T$ what really gives $B=B^T$.

Answer 3

Here's a nice counterexample: $$ \pmatrix{2&0\\0&1} \pmatrix{1&1\\1&1} \pmatrix{2&0\\0&1}^{-1} = \pmatrix{1&2\\1/2&1} $$