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What is the maximmal dimension of a vector subspace of $\mathcal{M}_n(\Bbb{R})$ formed by diagonalisable matrices $\mathcal{D}_n(\Bbb{R})$?

Attempt :

Let $\mathcal{S}_n(\Bbb{R})$ the set of symmetric matrices, wich is a subspace of $\mathcal{D}_n(\Bbb{R})$ of dimension $\frac{n(n+1)}{2}$ and denote $\mathcal{T}_n(\Bbb{R})$ the set of upper triangular matrice with zero diagonal , wich is a subspace of $\mathcal{D}_n(\Bbb{R})$ of dimension $\frac{n(n-1)}{2}$. Then $\mathcal{S}_n(\Bbb{R})$ and $\mathcal{M}_n(\Bbb{R})$ are in direct sum.

How can I continue ?

NB: I am also curious if we replace $\Bbb{R}$ by $\Bbb{C}$ ?

1 Answers1

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The dimension of a diagonalisable matrix subspace $V$ over any field is always bounded above by $n(n+1)/2$. This is because $V\cap\mathcal T_n=0$ and $V+\mathcal T_n\subseteq\mathcal M_n$, so that $\dim V\le\dim\mathcal M_n-\dim\mathcal T_n$.

So, for the real case, by considering the space of all symmetric matrices, we see that the maximal dimension is indeed $n(n+1)/2$. However, for complex matrices (or over any algebraically closed field of characteristic $p$ with $p=0$ or $p\ge n$), by Motzkin-Taussky theorem, those matrices in the diagonalisable subspace must commute. It follows that all matrices in the subspace are simultaneously diagonalisable and hence the maximal dimension is $n$.

Related: All linear combinations diagonalizable over $\mathbb C$ implies commuting.

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  • Thanks but I do not see why using all symmetric matrices we have the result ? –  Jul 01 '14 at 22:42
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    @Yass Huh? The matrix subspace formed by all real symmetric matrices has dimension $n(n+1)/2$, doesn't it? – user1551 Jul 01 '14 at 22:49
  • Sure. I just do not understand why because of this and your first claim then we can conclude that the maximal demension is $\frac{n(n+1)}{2}$, now Ì have understand. Thanks. Just a curious question now, why community wiki ? –  Jul 01 '14 at 22:52
  • This is just a summary of some results in the literature, but not something that I've cooked up. Hence community wiki. – user1551 Jul 01 '14 at 22:56