Show that V is an internal direct sum of the eigenspaces

[Karl Karlsson]

Homework Statement

If V = $M_{n,n}(\mathbb R)$ is the vectorspace of nxn matrices for some n and $L(A) = A^T$, show that V is an internal direct sum
of the eigenspaces and show that L is diagonalizable.

Relevant Equations

xxx

I was in an earlier problem tasked to do the same but when V = $M_{2,2}(\mathbb R)$. Then i represented each matrix in V as a vector $(a_{11}, a_{12}, a_{21}, a_{22})$ and the operation $L(A)$ could be represented as $L(A) = (a_{11}, a_{21}, a_{12}, a_{22})$. This method doesn't really work well when we talk about general $n\times n$ matrices. How would one go about to solve this?

Thanks in advance!

 

[member 587159]

A bunch of ideas:

Have you been able to show that $L$ is diagonalisable? Maybe can you find some conditions the eigenvalues must satisfy? For example, you have that $\lambda$ is an eigenvalue with eigenvector $A\in M_n(\mathbb{R})$ if $\lambda A =A^T$. Taking determinants, what conditions do you obtain? Does this help to find some eigenvectors? Can you find a basis of eigenvectors?

 

[fresh_42]

The standard equation to think about in such cases is to consider $\dfrac{1}{2}\left(A\pm A^\tau\right)$, i.e. to separate the matrices in a symmetric and an antisymmetric part.

 

[Karl Karlsson]

The standard equation to think about in such cases is to consider $\dfrac{1}{2}\left(A\pm A^\tau\right)$, i.e. to separate the matrices in a symmetric and an antisymmetric part.

Thanks, that helped me solve the problem