Applied Mathematics 1411A/B Lecture Notes - Lecture 19: If And Only If, Dot Product, Diagonalizable Matrix

• School: Western University
• Department: Applied Mathematics
• Course: Applied Mathematics 1411A/B
• Professor: Natalia Kiriushcheva

Document Summary

Recall: last day, we introduced the concept of diagonalizing a matrix. Theorem: if a is an nn matrix, then the following are equivalent: a is diagonalizable, a has n linearly independent eigenvectors. nn , the eigenvectors are in rn. Thus, b) implies that the: find n linearly independent eigenvectors of a, say, form the matrix p having, the matrix will then be diagonal with np as its column vectors. pp. 1 as its successive i is the eigenvalue corresponding to the. Remark: since a is eigenvectors form a basis for rn. Example: find the matrix p that diagonalizes a. diagonal entries, where eigenvector. Example: find the matrix p that diagonalizes b. , then nn matrix a has n distinct eigenvalues, then a is diagonalizable. kv are eigenvectors of a corresponding to the distinct eigenvalues. Definition: if eigenspace corresponding to number of times that. Examples: for the matrices below, determine the algebraic and geometric multiplicity of their eigenvalues.