Positive Definite Matrices eigenvalues

In summary, positive definite matrices have a special property where, if A is symmetric, A-λI is positive definite if and only if all eigenvalues of A are greater than λ. And if A-λI is negative definite, then all eigenvalues of A are less than λ. Using this result, it can be shown that all eigenvalues of the given matrix are between zero and eight. However, it is important to show your own attempt at a solution in order to receive help. Additionally, please make sure to post in the appropriate section for homework questions.
  • #1
angelz429
24
0
[SOLVED] Positive Definite Matrices

a) If A is Symmetric show that A-λI is positive definite if and only if all eigenvalues of A are >λ, and A-λI is negative definite if and only if all eigenvalues of A are <λ.

b) Use this result to show that all the eigenvalues of
[ 5 2 -1 0]
[ 2 5 0 1]
[-1 0 5 -2]
[ 0 1 -2 5]
are between zero and eight.
 
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  • #2
Didn't you read the files you were asked to when you registered?

You have now posted three consecutive questions (which look like homework and so should have been posted in the homework sections) without showing any work or any attermpt at a solution by yourself. You are required to show what you have done so we will know what kind of help you need.
 
  • #3
So I made a mistake, and I figured it out. It's not like I can delete them. I have already posted to homework, and gotten it solved. Thanks though.
 

What is a positive definite matrix?

A positive definite matrix is a square matrix in which all of its eigenvalues are positive. This means that the matrix is symmetric and all of its principal minors (determinants of submatrices) are positive.

Why are positive definite matrices important?

Positive definite matrices are important because they have many useful properties, such as being invertible and having unique positive definite square roots. They are also commonly used in optimization problems and in the study of quadratic forms.

How do you find the eigenvalues of a positive definite matrix?

The eigenvalues of a positive definite matrix can be found by solving the characteristic polynomial of the matrix, which is a polynomial equation of degree n (where n is the size of the matrix). Alternatively, they can also be found by using specialized algorithms such as the Cholesky decomposition.

What is the relationship between the eigenvalues of a positive definite matrix and its positive definite square root?

The eigenvalues of a positive definite matrix are the same as the eigenvalues of its positive definite square root. This means that the eigenvalues of the square root matrix are also positive, making it a positive definite matrix as well.

Can a positive definite matrix have complex eigenvalues?

No, a positive definite matrix can only have real and positive eigenvalues. This is because the definition of a positive definite matrix requires all of its eigenvalues to be positive real numbers.

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