Positive definite matrix and its eigenvalues

In summary, a positive definite matrix is a square matrix with all positive eigenvalues, indicating a unique positive solution to the equation Ax=b for any non-zero vector b. This can be determined by checking the signs of the eigenvalues or by satisfying the definition of positive definiteness. Positive definite matrices have applications in optimization problems, solving linear equations, and analyzing dynamic systems. The eigenvalues of a positive definite matrix are all positive and related in terms of their magnitudes. Non-square matrices cannot be positive definite as they do not have eigenvalues.
  • #1
retspool
36
0
I need to know if there is any relationship between the positive definite matrices and its eigenvalues

Also i would appreciate it if some one would also include the relationship between the negative definite matrices and their eigenvalues

Also can some also menthow the Gaussian elimination of the matrix help in determining the above

Best
 
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  • #2
what's your definition of positive definite?

try using it with an arbitrary eigenvector
 

What is a positive definite matrix?

A positive definite matrix is a square matrix where all of its eigenvalues are positive. This means that the matrix has a unique solution to the equation Ax = b for any non-zero vector b, and the solution is always positive.

How can I determine if a matrix is positive definite?

One way to determine if a matrix is positive definite is by checking the signs of its eigenvalues. If all of the eigenvalues are positive, then the matrix is positive definite. Another method is to check if the matrix satisfies the definition of positive definiteness, which states that for any non-zero vector x, xTAx > 0.

What are the applications of positive definite matrices?

Positive definite matrices are commonly used in optimization problems, such as in machine learning algorithms and in engineering designs. They are also useful in solving systems of linear equations and in analyzing the stability of dynamic systems.

How are the eigenvalues of a positive definite matrix related?

The eigenvalues of a positive definite matrix are all positive and are related to each other in terms of their magnitudes. The largest eigenvalue represents the maximum stretching or scaling factor of the matrix, while the smallest eigenvalue represents the minimum stretching or scaling factor. The intermediate eigenvalues represent the stretching or scaling factors in between.

Can a non-square matrix be positive definite?

No, a non-square matrix cannot be positive definite. This is because a non-square matrix does not have eigenvalues, and the definition of positive definiteness requires the existence of eigenvalues. Only square matrices can have eigenvalues and therefore be classified as positive definite.

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