# Matrix diagonalization theorems and counterexamples: reference-request.

I'm looking for exhaustive list of diagonalization theorems and counterexamples in linear algebra.

In this question I understand the question of matrix diagonalization very broadly: suppose we have some group $G$ of matrices, where $G$ is one of

1. $\operatorname{Gl}(n,\mathbb{R})$ - real invertible matrices,
2. $\operatorname{Gl}(n,\mathbb{C})$ - complex invertible matrices,
3. $\operatorname{O}(n)$ - orthogonal matrices,
4. $\operatorname{SO}(n)$ - special orthogonal matrices,
5. $\operatorname{U}(n)$, $\operatorname{SU}(n)$ - unitary and special unitary matrices,
6. generalization of 3-5: matrices with real or complex coefficients, satisfying either of $M C M^{-1} = C$ or $M C M^* = C$ for some fixed matrix $C$. This includes e.g. Lorentz group, group of symplectic matrices.

Also we have some set $S$ of matrices we want to diagonalize, where $S$ is one of

1. $M^n(\mathbb{R})$ - all real valued matrices,
2. $M^n(\mathbb{C})$ - all complex-valued matrices,
3. self-adjoint matrices over $\mathbb{R}$ or $\mathbb{C}$,
4. anti-symmetric matrices over $\mathbb{R}$,
5. normal matrices over $\mathbb{R}$ or $\mathbb{C}$.

I'm interested in the orbits of action of $G$ on $S$, where action is one of $s \mapsto gsg^{-1}$ ($s$ is interpreted as an endomorphism) and $s\mapsto gsg^*$ ($s$ is interpreted as a bilinear form): what is the "nicest" form of matrix, which we can find on any orbit? May be its not exactly diagonal, but Jordan, or made from $2\times 2$ blocks on the diagonal - all of these are examples of "nice" forms. The typical questions I would ask are

1. Is it true, that for a real matrix $m$, its diagonizability by $\operatorname{Gl}(n,\mathbb{C})$ (with the obtained diagonal matrix being real) implies the diagonizability by $\operatorname{Gl}(n,\mathbb{R})$ (as an endomorphism)? (Yes)
2. Is every (real) symmetric matrix diagonizable as a bilinear form by Lorentz transformations (in $\mathbb{R}^4$)?
3. Can every real antisymmetric form be transformed by the same Lorentz group to a matrix with blocks of the form $\left(\begin{smallmatrix} 0 & -\lambda \\ \lambda & 0 \end{smallmatrix}\right)$ on the diagonal?
4. Can any real matrix $m$ be transformed to a matrix with $1\times1$ and $2\times 2$ blocks on the diagonal by $SO(n)$? (No) What if you are allowed to have Jordan-like form, i.e. whenever two blocks on the diagonal are the same, one can have identity matrix on the diagonal next to it? (Still no by the dimension counting argument) Ok, what is the simplest shape one can hope for?

Note that two transformation types coincide for $SO(n)$, $O(n)$, $U(n)$, $SU(n)$, so I don't need to specify one in the question #3.

Is there any extensive reference I can look up all these questions and future similar questions?

• The fundamental question here is to broad for this forum. As for the reference request, very few linear algebra books provide tables of all combinations of question one might ask with their answers (the theory just does not work that way). – Marc van Leeuwen Aug 29 '15 at 5:21