Positive Definite Matrices eigenvalues

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SUMMARY

The discussion focuses on the properties of positive definite matrices, specifically regarding eigenvalues. It establishes that for a symmetric matrix A, the matrix A-λI is positive definite if all eigenvalues of A are greater than λ, and negative definite if all eigenvalues are less than λ. The example matrix provided demonstrates that all eigenvalues lie between zero and eight, confirming the theoretical assertions made in the discussion.

PREREQUISITES
  • Understanding of symmetric matrices
  • Knowledge of eigenvalues and eigenvectors
  • Familiarity with matrix operations, specifically matrix subtraction and scalar multiplication
  • Basic concepts of positive and negative definiteness in linear algebra
NEXT STEPS
  • Study the properties of symmetric matrices in linear algebra
  • Learn about the spectral theorem and its implications for eigenvalues
  • Explore numerical methods for calculating eigenvalues of matrices
  • Investigate applications of positive definite matrices in optimization problems
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Students and professionals in mathematics, particularly those studying linear algebra, as well as researchers and practitioners working with optimization and matrix analysis.

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[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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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.
 
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.
 

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