Linear Algebra · Lesson 06
Bring the chapter together: find eigenvalues and eigenvectors, compare the two multiplicities, decide whether a matrix diagonalises, and use the eigenvalues to predict powers. No worked example sits above the answers.
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Sign in to save progressBring the chapter together: find eigenvalues and eigenvectors, compare the two multiplicities, decide whether a matrix diagonalises, and use the eigenvalues to predict powers. No worked example sits above the answers.
A doctor reads a few vital signs and knows a great deal about the patient. The eigenvalues are the vital signs of a matrix.
Decide first what is asked. Eigenvalues: the roots of , or the diagonal of a triangular matrix. Eigenvectors: the null space of . Diagonalisable: enough eigenvectors for a basis. Powers and the long run: and the sizes of the eigenvalues.
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What is the larger eigenvalue of ?
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Which is an eigenvector of for the eigenvalue ?
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A matrix has eigenvalues , and , counted with multiplicity. What is its determinant?
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The eigenvalues of an upper triangular matrix are its diagonal entries.
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Which matrix is not diagonalisable?
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with and . What is the entry in row 1, column 2 of ?
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If is an eigenvalue of an invertible matrix , then is an eigenvalue of .
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A linear map of the plane sends every vector on the line to itself and every vector on the line to its negative. What can be said about it?
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is an eigenvector of with eigenvalue . What is the eigenvalue of for the same ?
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Every 2 by 2 matrix with real entries has a real eigenvalue.