Question: What is the nature of the given squre Matrix if the eigen values are complex numbers?

Good morning sir,

In  general, there are five ways to decide the nature of a given square matrix based on the signs of the eigenvalues, for example

1. If the eigenvalues are positive then the nature is positive definite.
2. If the eigenvalues are negative then the nature is negative definite.
3. If the eigen values are nonnegative then the nature is Positive semidefinite.
4. If the eigenvalues are nonpositive then the nature is negative semi definite.
5. If the eigenvalues are positive and negative then the nature is indefinite.

My query is that if the eigenvalues are complex numbers then what is the nature of the square matrix?

The positive definite matrices are generally diagonalizable.

I request you to comment on the following matrix A whether it is positive definite, if so express it in the form

D=Q^tAQ where Q is the orthogonal matrix and D is the diagonal matrix.

The matrix A is been uploaded as an attachment. Please find the enclosed file.

With thanks & Regards

M.Anand

Assistant Professor in Mathematics

SR International Institute of Technology,

Hyderabad, Andhra Pradesh, INDIA.