Linear Algebra: Hermitian Matrices

In summary, the conversation revolves around questions regarding Hermitian 2x2 matrices and their eigenvectors. The answer to both questions 1 and 2 is 'yes', and it is also confirmed that a Hermitian nxn matrix will always have n (not necessarily distinct) eigenvalues. It is also mentioned that 'Dick' is a common nickname for 'Richard', with no clear origin for the letter 'D'.
  • #1
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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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  • #2
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?
 
  • #3
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?
 
  • #4
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.
 
  • #5
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): [Broken]

"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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1. What is a Hermitian matrix?

A Hermitian matrix is a square matrix that is equal to its own conjugate transpose. In other words, if A is a Hermitian matrix, then A is equal to A*, where A* is the conjugate transpose of A.

2. How is a Hermitian matrix different from a symmetric matrix?

A Hermitian matrix is a generalization of a symmetric matrix in complex numbers. While a symmetric matrix is equal to its own transpose, a Hermitian matrix is equal to its own conjugate transpose.

3. What are the properties of Hermitian matrices?

Some key properties of Hermitian matrices include:
- All eigenvalues are real
- Eigenvectors corresponding to distinct eigenvalues are orthogonal
- The trace of a Hermitian matrix is equal to the sum of its eigenvalues
- All submatrices formed by selecting rows and columns of a Hermitian matrix are also Hermitian matrices

4. How are Hermitian matrices used in linear algebra?

Hermitian matrices are used in linear algebra for a variety of applications, including:
- Diagonalization of matrices
- Quadratic forms
- Inner product spaces
- Orthonormal bases
- Unitary matrices
- Spectral theorem

5. Can a non-square matrix be a Hermitian matrix?

No, a non-square matrix cannot be a Hermitian matrix. Hermitian matrices are always square matrices with the same number of rows and columns.

Suggested for: Linear Algebra: Hermitian Matrices

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