Showing that something is in eigenspace Watch

Valentas
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Report Thread starter 5 years ago
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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.
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ghostwalker
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(Original post by Valentas)
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.
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Valentas
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(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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