Linear Algebra: Hermitian Matrices

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Homework Help Overview

The discussion revolves around properties of Hermitian matrices, specifically focusing on eigenvalues and eigenvectors of 2x2 matrices. The original poster poses questions regarding the span of eigenvectors and their relationship between two distinct Hermitian matrices.

Discussion Character

  • Conceptual clarification, Assumption checking

Approaches and Questions Raised

  • The original poster questions whether the eigenvectors of Hermitian matrices span R2 and if the eigenvectors of two different Hermitian matrices span the same space. Participants discuss the linear independence of eigenvectors corresponding to distinct eigenvalues.

Discussion Status

Some participants confirm the original poster's understanding of the properties of eigenvectors related to distinct eigenvalues. There is an exploration of the implications of these properties, although the discussion also diverges into unrelated topics.

Contextual Notes

The original poster seeks confirmation on specific properties of Hermitian matrices, including the nature of eigenvalues and the span of eigenvectors, while also including a side note about the nickname "Dick" for Richard.

Niles
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Homework Statement


Hi all.

Let's say that I have a Hermitian 2x2 matrix A with two distinct eigenvalues, and thus two eigenvectors.

Question 1: What space is it they span? Is it R2?

Now let us say I have another Hermitian 2x2 matrix B with two distinct eigenvalues, and thus two eigenvectors.

Question 2: Do the eigenvectors of B span the same space as the eigenvectors of A?

Thanks in advance.


Niles.
 
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If A has two distinct eigenvalues, then the corresponding eigenvectors must be linearly independent, right? Isn't this true regardless of whether A is hermitian or not?
 
Yes, you are correct. A minor detail I was not aware of. Can you confirm me in that the answers to question 1 and 2 are "yes"?

By the way, what name is "Dick" short for? I mean, Richard Feynman was called "Dick Feynman", but they never explain in any of the books (at least the ones I've read) what it is short for.

Thanks in advance.

Regards,
Niles.

EDIT: I just need to get this confirmed. A Hermitian nxn matrix will always have n (not necessarily distinct) eigenvalues, since it is diagonalizable, right?
 
Yes, the answers to A and B are 'yes'. And yes, a hermitian nxn matrix has n (not necessarily distinct) eigenvalues. In the sense that it's characteristic equation has n real linear factors. 'Dick' is a nickname for 'Richard' (just like 'Rick'). I have no idea where the 'D' came from, it's just what people call me.
 
Dick said:
Yes, the answers to A and B are 'yes'. And yes, a hermitian nxn matrix has n (not necessarily distinct) eigenvalues. In the sense that it's characteristic equation has n real linear factors.
Great. Thanks.
Dick said:
'Dick' is a nickname for 'Richard' (just like 'Rick'). I have no idea where the 'D' came from, it's just what people call me.
Ahh, I see. This I found from Wikipedia (http://en.wikipedia.org/wiki/Richard):

"The first or given name Richard comes from the Germanic elements "ric" (ruler, leader, king) and "hard" (strong, brave)." Interesting.

Thanks again.
 
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