Solved Problems for Chapter 8

If is an invertible matrix, compare the eigenvalues of and . More generally, for an arbitrary integer, compare the eigenvalues of and .

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If is an matrix and is a nonzero constant, compare the eigenvalues of and .

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Let be invertible matrices which commute. That is, . Suppose is an eigenvector of . Show that then must also be an eigenvector for .

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Suppose is an matrix and it satisfies for some a positive integer larger than 1. Show that if is an eigenvalue of then equals either 0 or .

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Show that if and , then whenever are scalars, Does this imply that is an eigenvector? Explain.

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Find the eigenvalues and eigenvectors of the matrix One eigenvalue is

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Find the eigenvalues and eigenvectors of the matrix One eigenvalue is .

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Is it possible for a nonzero matrix to have only as an eigenvalue?

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Let  be the linear transformation which reflects vectors about the axis. Find a matrix for and then find its eigenvalues and eigenvectors.

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Let be the linear transformation which reflects all vectors in through the plane. Find a matrix for and then obtain its eigenvalues and eigenvectors.

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Find the eigenvalues and eigenvectors of the matrix One eigenvalue is Diagonalize if possible.

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Find the eigenvalues and eigenvectors of the matrix One eigenvalue is Diagonalize if possible.

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Find the eigenvalues and eigenvectors of the matrix One eigenvalue is Diagonalize if possible.

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If and has any of the following properties, show that has the same property.
(a)
A is Idempotent, that is .
(b)
A is Nilpotent, that is for some .

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(c)
A is Invertible.

Bibliography

These problems come from Chapter 7 of Ken Kuttler’s A First Course in Linear Algebra. (CC-BY)

Ken Kuttler, A First Course in Linear Algebra, Lyryx 2017, Open Edition, pp. 359–401.

2024-09-06 23:26:00