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Practice Question: Eigenvalues of Triangular Matrices

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Wize University Linear Algebra Textbook > Eigenvalues and Eigenvectors (Spectral Theory)

Properties

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Prove that any element on the main diagonal of a triangular matrix is an eigenvalue.
More Properties Questions:
133 - FML 3 - 18.1W e.g. 19
If   ⁣A‾\bcb{\ul{A}} be a 3×3\bcb{3 \times 3} matrix where trace(  ⁣A‾)=−1\bcb{\text{trace}(\ul{A}) = -1} and det(  ⁣A‾)=−1\bcb{\det{\ul{A}} = -1}, then the possible eigenvalues of   ⁣A‾\bcb{\ul{A}} are:
133 - FML 3 - 18.1W e.g. 18
If   ⁣A‾\bcb{\ul{A}} is a 4×4\bcb{4 \times 4} matrix with eigenvalues λ1=λ2=λ3=1\bcb{\lambda_1 = \lambda_2 = \lambda_3 = 1} and λ4=2\bcb{\lambda_4 = 2}, which of the following statements are true?
133 - FML 3 - 18.1W e.g. 70.5
Suppose 5\bcb{5} and −2\bcb{-2} are eigenvalues of the matrix   ⁣A‾\bcb{\ul{A}}, which of the following statements must be true?
det(  ⁣A‾−5  ⁣I‾)=det(  ⁣A‾+2  ⁣I‾)\bcb{\det{\ul{A} - 5\ul{I}} = \det{\ul{A} + 2 \ul{I}}}.
133 - FML 3 - 18.1W e.g. 70.4
Suppose 5\bcb{5} and −2\bcb{-2} are eigenvalues of the matrix   ⁣A‾\bcb{\ul{A}}, which of the following statements must be true?
trace(  ⁣A‾−5  ⁣I‾)=trace(  ⁣A‾+2  ⁣I‾)\bcb{\text{trace}(\ul{A} - 5\ul{I}) = \text{trace}(\ul{A} + 2\ul{I})}.
133 - FML 3 - 18.1W e.g. 70.3
Suppose 5\bcb{5} and −2\bcb{-2} are eigenvalues of the matrix   ⁣A‾\bcb{\ul{A}}, which of the following statements must be true?
det(  ⁣A‾−5  ⁣I‾)=0\bcb{\det{\ul{A} - 5\ul{I}} = 0}, and   ⁣A‾v⃗=−2v⃗\bcb{\ul{A} \ol{v} = -2 \ol{v}} has only the trivial solution.
133 - FML 3 - 18.1W e.g. 70.2
Suppose 5\bcb{5} and −2\bcb{-2} are eigenvalues of the matrix   ⁣A‾\bcb{\ul{A}}, which of the following statements must be true?
A−5I\bcb{A-5I} is invertible and det(  ⁣A‾+2  ⁣I‾)=0\bcb{\det{\ul{A} + 2\ul{I} } = 0}.
133 - FML 3 - 18.1W e.g. 70.1
Suppose 5\bcb{5} and −2\bcb{-2} are eigenvalues of the matrix   ⁣A‾\bcb{\ul{A}}, which of the following statements must be true?
A−5I\bcb{A - 5I} is not invertible and there is a nontrivial solution to   ⁣A‾v⃗=−2v⃗\bcb{\ul{A} \ol{v} = -2 \ol{v}}.
Properties
Practice: Similar Matrices
Consider the matrix A=[−1421]A= \left[ \begin{array}{rrr} -1 & 4\\ 2 & 1 \end{array} \right].