Howtosignalprocessing
- How do you find the specific eigenvalues of a matrix?
- Are eigenvectors unique?
- What happens to eigenvalues when you add matrices?
- How are the eigenvalues of a matrix related to the trace of the matrix?
How do you find the specific eigenvalues of a matrix?
In order to determine the eigenvectors of a matrix, you must first determine the eigenvalues. Substitute one eigenvalue λ into the equation A x = λ x—or, equivalently, into ( A − λ I) x = 0—and solve for x; the resulting nonzero solutons form the set of eigenvectors of A corresponding to the selectd eigenvalue.
Are eigenvectors unique?
This is a result of the mathematical fact that eigenvectors are not unique: any multiple of an eigenvector is also an eigenvector! Different numerical algorithms can produce different eigenvectors, and this is compounded by the fact that you can standardize and order the eigenvectors in several ways.
What happens to eigenvalues when you add matrices?
The eigenvalues of a sum of matrices C=A+B equal the sum of their eigenvalues, that is, c_n = a_n+b_n, only in the most special of cases. A and B diagonal is one such case.
How are the eigenvalues of a matrix related to the trace of the matrix?
The trace of a matrix A, designated by tr(A), is the sum of the elements on the main diagonal. The sum of the eigenvalues of a matrix equals the trace of the matrix.
