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Consider the matrix A =

Consider the matrix A =

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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2 Consider the matrix A = 3 4. Show that p(A)v = p(c)v holds in general, if v is an eigenvector of A for the eigenvalue c. 5. Assume now that v1, v2, ..., Un are eigenvectors to different eigenvalues c1, C2, ..., Cn. We want to show that these eigenvectors are linearly indepdendent. To do so, write down a linear combination of the form v = bv1 + bzvzt... +bn Un = 0 and show that all coefficient by are zero. To show, for example, that bị is zero, you can consider the polynomial P1 (z) = (z – c2)(r – c3)... (z - Cn) and evalutate pi (A)v. %3D %3D
Transcribed Image Text:2 Consider the matrix A = 3 4. Show that p(A)v = p(c)v holds in general, if v is an eigenvector of A for the eigenvalue c. 5. Assume now that v1, v2, ..., Un are eigenvectors to different eigenvalues c1, C2, ..., Cn. We want to show that these eigenvectors are linearly indepdendent. To do so, write down a linear combination of the form v = bv1 + bzvzt... +bn Un = 0 and show that all coefficient by are zero. To show, for example, that bị is zero, you can consider the polynomial P1 (z) = (z – c2)(r – c3)... (z - Cn) and evalutate pi (A)v. %3D %3D
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