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