5.1 - MATH1850U/2050U: Chapter 5 1 EIGENVALUES AND...

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MATH1850U/2050U: Chapter 5 1 EIGENVALUES AND EIGENVECTORS Eigenvalues and Eigenvectors (5.1; pg. 295) Definition: If A is an n n matrix, then a nonzero vector x in R n is called an eigenvector of A if x A is a scalar multiple of x ; that is x x A for some scalar . The scalar is called an eigenvalue of A , and x is said to be an eigenvector of A corresponding to . Example: Question: But how would we find eigenvalues/eigenvectors in the first place? Definition: The equation 0 ) det( A I is called the characteristic equation of A . The scalars that satisfy this equation are the eigenvalues of A . When expanded, the characteristic equation is a polynomial in , and is called the characteristic polynomial of A .
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Chapter 5 2 Example: Find the eigenvalues of the matrix below. Theorem (Eigenvalues of a triangular matrix): If A is an n n matrix (upper triangular, lower triangular, or diagonal), then the eigenvalues of A are the entries on the main diagonal of A . Example: Find the eigenvalues of the matrix below. Example: Remark: The eigenvalues of a matrix may be complex numbers. Example:
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5.1 - MATH1850U/2050U: Chapter 5 1 EIGENVALUES AND...

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