Technological University of the Philippines – Advance Mathematics – Electronics Engineering

Eigen Values

WHAT ARE EIGEN VALUES?

Technological University of the Philippines – Advance Mathematics – Electronics Engineering

Eigen Values•Are a special set of scalars

associated with a linear system of

equations (i.e., a matrix equation).

Technological University of the Philippines – Advance Mathematics – Electronics Engineering

Eigen Values• They are sometimes also known as

• characteristic roots, characteristic values

(Hoffman and Kunze 1971)

• proper values, or latent roots (Marcus and Minc

1988, p. 144).

Technological University of the Philippines – Advance Mathematics – Electronics Engineering

Eigen Values

HOW TO DETERMINE THE EIGEN VALUES?

Technological University of the Philippines – Advance Mathematics – Electronics Engineering

Eigen Values•Letting A be a kxk square matrix

Technological University of the Philippines – Advance Mathematics – Electronics Engineering

Eigen Values•With eigenvalue , then the corresponding eigenvectors satisfy

Technological University of the Philippines – Advance Mathematics – Electronics Engineering

Eigen Values•With is equivalent to the homogenous system

Technological University of the Philippines – Advance Mathematics – Electronics Engineering

Eigen Values•Equation can be written compactly as

Technological University of the Philippines – Advance Mathematics – Electronics Engineering

Eigen Values•Where I is the identity matrix. As shown in Cramer’s rule, a linear system of equations has nontrivial solutions if the determinant vanishes, so the solutions of the equation is given by

Technological University of the Philippines – Advance Mathematics – Electronics Engineering

Eigen Values

•This equation is known as the characteristic equation of A, and the left hand side is known as the characteristic polynomial.

Technological University of the Philippines – Advance Mathematics – Electronics Engineering

Eigen Values

EXAMPLES?

Technological University of the Philippines – Advance Mathematics – Electronics Engineering

Eigen Values•Find the eigenvalues of the given matrix

Technological University of the Philippines – Advance Mathematics – Electronics Engineering

Eigen Values•Solution:

Technological University of the Philippines – Advance Mathematics – Electronics Engineering

Eigen Values•Solution:

Technological University of the Philippines – Advance Mathematics – Electronics Engineering

Eigen Values•Solution:

Technological University of the Philippines – Advance Mathematics – Electronics Engineering

Eigen Values•Solution:

Technological University of the Philippines – Advance Mathematics – Electronics Engineering

Eigen Values•Find the eigenvalues of the given matrix

Technological University of the Philippines – Advance Mathematics – Electronics Engineering

Eigen Values•Solution:

Technological University of the Philippines – Advance Mathematics – Electronics Engineering

Eigen Values•Solution:

Technological University of the Philippines – Advance Mathematics – Electronics Engineering

Eigen Values• Solution:

Technological University of the Philippines – Advance Mathematics – Electronics Engineering

Eigen Values• Solution: