I am not sure if a square matrix A is symetric ($\displaystyle
$$A = {A^T}$$$) when we have the relation $\displaystyle $$AP{A^{ - 1}} = {A^T}P{({A^T})^{ - 1}}$$
$. Has anyone an idea how to prove this?
Thanks Ulrich
I think you are after something like this
Symmetric Matrices
Yes, but I think it is not possible to transform the first equation into the second because the matrices are supposed to not commute. Maybe it should be done by supposing that the first equation is also valid for
$\displaystyle $$A \ne {A^T}$$ $
and then arrive at some contradiction. Or equivalently puting
$\displaystyle $$A = {A^T}B$$$, from which follows
$\displaystyle $$AP{A^{ - 1}} = {A^T}BP{B^{ - 1}}{({A^T})^{ - 1}}$$$
and by comparision
$\displaystyle $$BP{B^{ - 1}} = P$$$.
Does this imply that B is the identity matrix? If yes, why? It's some time ago when I took courses in linear algebra...
I found out that $\displaystyle $$BP{B^{ - 1}} = P$$$ does not necessarily mean that B is the identity matrix. If for instance
$\displaystyle P=\left(
\begin{array}{ccc}
0 & 0 & 1 \\
1 & 0 & 0 \\
0 & 1 & 0
\end{array}
\right)$
and
$\displaystyle B=\left(
\begin{array}{ccc}
i & f & h \\
h & i & f \\
f & h & i
\end{array}
\right)$
then this is true but such solutions only yield $\displaystyle A = 0$ for $\displaystyle $$A = {A^T}B$$$ so A must be symmetric.
Counterexample:
$\displaystyle A=\begin{bmatrix}1 &2\\ 3 &4\end{bmatrix}.$ Clearly, $\displaystyle A\not=A^{T}.$
Let $\displaystyle P=\begin{bmatrix}-3 &-4\\ 1 &2\end{bmatrix}.$
Then, by direct computation, $\displaystyle APA^{-1}=A^{T}P(A^{T})^{-1},$ but $\displaystyle A\not=A^{T}.$
Also note that this $\displaystyle P$ does not commute with $\displaystyle A.$
For reference,
$\displaystyle A^{-1}=\begin{bmatrix}-2 &1\\ 3/2 &-1/2\end{bmatrix},$ and, of course,
$\displaystyle (A^{T})^{-1}=(A^{-1})^{T}.$
So I think this answers your hypothesis in the negative.
Hi all,
I had some time recently and also found, as a "by-product", the solution to this problem. In fact the matrices
$\displaystyle A=\begin{bmatrix}1 &2\\ 3 &4\end{bmatrix}$ and
$\displaystyle P=\begin{bmatrix}-3 &-4\\ 1 &2\end{bmatrix}$
do not commute but are nevertheless special because $\displaystyle BP = PB$ where $\displaystyle B=(A^{T})^{-1}A$ (this is also true for the 3-dimensional matrices I mentionned before).
From this follows that $\displaystyle BP{B^{ - 1}} = P$ with B not necessarily being the identity matrix. So if one imposes that B and P do not commute, then $\displaystyle APA^{-1}=A^{T}P(A^{T})^{-1}$ implies that A is symmetric.
I also asked earlier how it is possible to show from $\displaystyle BP{B^{ - 1}} = P$ that B is the identity matrix. This can be done as follows:
Given P and Q two arbitray matrices, we want to determine another matrix B such that $\displaystyle BP {B^{ - 1}} = Q$ is verified. From some theorem (sorry forgot the name) there is a matrix E such that Q can be diagonalised. In other words, we get $\displaystyle EQ{E^{ - 1}} = D$ where E is the matrix formed by the eigenvectors of Q, let's say $\displaystyle E=M(EV Q)$. D is the diagonal matrix with the eigenvalues of Q. This also yields
$\displaystyle E BP {B^{ - 1}}{E^{ - 1}} = (EB)P {(EB)^{ - 1}} = D$.
So EB is the matrix formed with the eigenvectors of P, that is $\displaystyle EB=M(EV P)$, which yields $\displaystyle B=E^{ - 1}}M(EV P)=M^{ - 1}}(EV Q)M(EV P)$.
This is why, if we have $\displaystyle P=Q$ like above, B must be the identity matrix.