Question

Verify that u=[1,13]^{T} is an eigenvector of the matrix
[[ -8,1],[-13,6]]. Find the corresponding eigenvalue lambda.

Answer #1

v is an eigenvector with eigenvalue 5 for the invertible matrix
A. Is v an eigenvector for A^-2? Show why/why not.

Let A be an n × n matrix and let x be an eigenvector of A
corresponding to the eigenvalue λ . Show that for any positive
integer m, x is an eigenvector of Am corresponding to the
eigenvalue λ m .

Given Eigenvalue 3, -2. Respective Eigenvector V1 = [1 1], V2=
[1 -1]. Find the matrix A

find the eigenvalues of the following matrix. then find the
corresponding eigenvector(s) of one ofthose eigenvalues (pick your
favorite).
1 -2 0
-1 1 -1
0 -2 1

The matrix A has an eigenvalue λ with an algebraic multiplicity
of 5 and a geometric multiplicity of 2. Does A have a generalised
eigenvector of rank 3 corresponding to λ? What about a generalised
eigenvector of rank 5?

Use the power method to determine the highest eigenvalue and
corresponding eigenvector for
[ (2-λ) 8 10; 8 (4-λ) 5; 10 5 (7-λ)]
(By hand with handy formulas please)

Complex Eigenstuff Compute the eigenvalues and eigenvectors for
the given matrix A. List the eigenvalues so the first one has
negative imaginary part. Write the corresponding eigenvectors in
the form [u+iv1]. If there is only one eigenvector, leave the
entries for the second eigenvalue and eigenvector blank. A=[4 -3 3
4]

Let M=[[116,−48],[−48,44]]. Notice that 20 is an eigenvalue of
M. Let U be an orthogonal matrix such that (U^−1)(M)(U) is
diagonal, the first column of U has positive entries, and det(U)=1.
Find (√20)⋅U.

Find the 3 * 3 matrix A corresponding to orthogonal projection
onto the solution
space of the system below.
2x + 3y + z = 0;
x - 3y - z = 0:
Your solution should contain the following information: (a) The
eigenvector(s) of
A that is (are) contained in the solution space; (b) The
eigenvector(s) of A that
is (are) perpendicular to the solution space; (c) The corresponding
eigenvalues for
those eigenvectors.

Let lambda be an eigenvalue of A in X'=AX of
multiplicity n. Find the general solution of
dx/dt = -5x +3y
dy/dt = -3x + y
Use eigenvalues, eigenvectors, and coefficients in
your method.

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