Linear Algebra: Hermitian Matrices

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SUMMARY

The discussion centers on Hermitian matrices, specifically 2x2 matrices A and B with distinct eigenvalues and their corresponding eigenvectors. It is established that the eigenvectors of a Hermitian matrix span R² when the matrix has two distinct eigenvalues, confirming their linear independence. Additionally, it is confirmed that a Hermitian nxn matrix will always have n eigenvalues, which may not necessarily be distinct, and its characteristic equation will have n real linear factors.

PREREQUISITES
  • Understanding of Hermitian matrices
  • Knowledge of eigenvalues and eigenvectors
  • Familiarity with linear independence concepts
  • Basic principles of diagonalization in linear algebra
NEXT STEPS
  • Study the properties of Hermitian matrices in detail
  • Learn about the spectral theorem for Hermitian matrices
  • Explore the implications of eigenvalues in quantum mechanics
  • Investigate the diagonalization process for various matrix types
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Students of linear algebra, mathematicians, and anyone interested in the properties of Hermitian matrices and their applications in various fields such as physics and engineering.

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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