Showing that something is in eigenspace

Let's say we have matrices X and Z which are both n x n matrices.

They have a property such that XZ = ZX

Let

\lambda

be an eigenvalue of X and

E_\lambda

it's eigenspace

1) Imagine that v

\not=

0 is element in

E_\lambda

Show that Zv is also element of

E_\lambda

Any help would be great.

Let's say we have matrices X and Z which are both n x n matrices.

They have a property such that XZ = ZX

Let

\lambda

be an eigenvalue of X and

E_\lambda

it's eigenspace

1) Imagine that v

\not=

0 is element in

E_\lambda

Show that Zv is also element of

E_\lambda

Any help would be great.

Well what does v as an element of

E_\lambda

tell you about its interaction with X?

And then how do you show Zv is an element of

E_\lambda

? What does it mean?

It drops out very quickly.

(edited 10 years ago)

Original post by ghostwalker

Well what does v as an element of

E_\lambda

tell you about its interaction with X?

And then how do you show Zv is an element of

E_\lambda

? What does it mean?

It drops out very quickly.

Thank you for hints. I managed to solve this. It was dead simple. I just needed to re-read the chapter and then apply knowledge from there.

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