$\DeclareMathOperator\Sym{Sym}\DeclareMathOperator\O{O}\DeclareMathOperator\GL{GL}$Let $k$ be an algebraically closed field of characteristic 0 (it can even be $\mathbb{C}$ if you like), and let $n\in\mathbb{N}$. Let $\Sym(n)$ denote the space of $n\times n$ symmetric matrices with entries in $k$. Let $\O(n)\subseteq \GL(n)$ denote the orthogonal group, that is all $n\times n$ matrices $A$ with entries in $k$ such that $AA^t=A^tA=I_n$.

Then $\O(n)$ acts on $\Sym(n)$ by conjugation. I would like nice representatives for the orbits of this action.

Let $B\in \Sym(n)$. If $B$ is diagonalizable, then it is a theorem that it can be diagonalized by an orthogonal matrix, so all the 'semisimple orbits' have a diagonal matrix as a representative.

So my question is about when $B$ is not diagonalizable — what is a 'nice' form we can put it in under orthogonal conjugation? Seeking explanations or references.

Thank you!

$\DeclareMathOperator\Sym{Sym}\DeclareMathOperator\O{O}\DeclareMathOperator\GL{GL}$Let $k$ be an algebraically closed field of characteristic 0 (it can even be $\mathbb{C}$ if you like), and let $n\in\mathbb{N}$. Let $\Sym(n)$ denote the space of $n\times n$ symmetric matrices with entries in $k$. Let $\O(n)\subseteq \GL(n)$ denote the orthogonal group, that is all $n\times n$ matrices $A$ with entries in $k$ such that $AA^t=A^tA=I_n$.

Then $\O(n)$ acts on $\Sym(n)$ by conjugation. I would like nice representatives for the orbits of this action.

Let $B\in \Sym(n)$. If $B$ is diagonalizable, then it is a theorem that it can be diagonalized by an orthogonal matrix, so all the 'semisimple orbits' have a diagonal matrix as a representative.

So my question is about when $B$ is not diagonalizable — what is a 'nice' form we can put it in under orthogonal conjugation? (For an example of a non-diagonalizable symmetric matrix, see https://math.stackexchange.com/questions/1658393/why-a-complex-symmetric-matrix-is-not-diagonalizible.)

Seeking explanations or references. Thank you!

Let $k$ be an algebraically closed field of characteristic 0 (it can even be $\mathbb{C}$ if you like), and let $n\in\mathbb{N}$. Let $Sym(n)$ denote the space of $n\times n$ symmetric matrices with entries in $k$. Let $O(n)\subseteq GL(n)$ denote the orthogonal group, that is all $n\times n$ matrices $A$ with entries in $k$ such that $AA^t=A^tA=I_n$.

Then $O(n)$ acts on $Sym(n)$ by conjugation. I would like nice representatives for the orbits of this action.

Let $B\in Sym(n)$. If $B$ is diagonalizable, then it is a theorem that it can be diagonalized by an orthogonal matrix, so all the 'semisimple orbits' have a diagonal matrix as a representative.

So my question is about when $B$ is not diagonalizable---what is a 'nice' form we can put it in under orthogonal conjugation? Seeking explanations or references.

Thank you!

$\DeclareMathOperator\Sym{Sym}\DeclareMathOperator\O{O}\DeclareMathOperator\GL{GL}$Let $k$ be an algebraically closed field of characteristic 0 (it can even be $\mathbb{C}$ if you like), and let $n\in\mathbb{N}$. Let $\Sym(n)$ denote the space of $n\times n$ symmetric matrices with entries in $k$. Let $\O(n)\subseteq \GL(n)$ denote the orthogonal group, that is all $n\times n$ matrices $A$ with entries in $k$ such that $AA^t=A^tA=I_n$.

Then $\O(n)$ acts on $\Sym(n)$ by conjugation. I would like nice representatives for the orbits of this action.

Let $B\in \Sym(n)$. If $B$ is diagonalizable, then it is a theorem that it can be diagonalized by an orthogonal matrix, so all the 'semisimple orbits' have a diagonal matrix as a representative.

So my question is about when $B$ is not diagonalizable — what is a 'nice' form we can put it in under orthogonal conjugation? Seeking explanations or references.

Thank you!