Is there a quick way of seeing if a matrix's eigenvalues are real or imaginary?

Question

What are some properties of matrices that imply real/imag eigenvalues? Is there a way to check just by inspection without computing the determinant?

Answer 1

A simple way to check it is that if the matrix is symmetric (that is, it is equal to its transpose, or in other words, it is symmetric around its main diagonal) then the eigenvalues HAVE TO be real.

If the matrix is skew-symmetric (that is, it is equal to minus one times its transpose, or in other words, all elements of the matrix are minus one times the corresponding elements on the other side of the main diagonal) then the eigenvalues HAVE TO be purely imaginary.

However, these are only sufficient conditions for real and purely imaginary eigenvalues, and they are not necessary. I do not know about necessary and sufficient conditions, so in the general case, you need to compute the determinant or use elimination.

Answer 2

A is a matrix; so that Ax = Lx and (A - L)x = 0; where L are the eigenvalues and Lx are the eigenvectors.

I don't see anything helpful in the definition, but you might find something in the trace of A to come up with the characteristic polynomial. But you need det(A) for part of that polynomial.

How about doing row column operations to reduce A to a diagonal matrix; so that the diagonal elements are the eigenvalues? By examining the A matrix elements, you might be able to see if any operations might result in taking the sqrt of a negative number. For one thing, a strictly positive A, where all elements>0 will never result in a negative eigenvalue, let alone an imaginary one.

Answer 3

Since the determination of eigenvalues for an n x n matrix is equivalent to finding the roots of an nth order polynomial, it certainly seems that there is no simple way of predicitng whether the eigenvalues of a general matrix are real or imaginary any more than you can predict the same about the roots of the polynomial.

Answer 4

Normaly if the matrix contains a negative then you cant find any real eigenvalues but personaly id stick to just working the determinant for small matrices. But if you have large positives in the same row or colum as you negative (i forget which) then your end matrix wont have negative, just stick to the determinant for now i am.