Applied Mathematics 1411A/B Chapter 5.1.3: Applied Mathematics 1411A/B Chapter 5.1.: Applied Mathematics 1411A/B Chapter 5.1: Applied Mathematics 1411A/B Chapter 5.: Section 5.1.3

• School: Western University
• Department: Applied Mathematics
• Course: Applied Mathematics 1411A/B
• Professor: Bogdan Tudose

Document Summary

If a is an n x n triangular matrix (upper triangular, lower triangular, or diagnol), then the eigenvalues of a are the entries on the main diagnol of a. That makes inspection a hell of a lot easier! If a is an n x n matrix, then the following statements are equivalent. a) b) c) d) Is a solution of the characteristic equation det( i - a) = 0. The system of equations ( i - a)x = 0 has nontrivial solutions. There is a nonzero vector x such that ax = x. A square matrix a is invertible if and only if = 0 is not an eigenvalue of a. This has to do with the theory of how the determinant has to be 0. So far we"ve only looked into eigenvalues and eigenvectors in the context of matrices and linear operators in rn. Let"s extend this concept to general vector spaces.