Question

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

Answer #1

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.

Find the eigenvalues and the eigenvectors corresponding to them
of the matrix
-2
1
3
0
-2
6
0
0
4

Find the characteristic equation and the eigenvalues (and
corresponding eigenvectors) of the matrix. 0 −3 5 −4 4 −10 0 0
4
(a) the characteristic equation (b) the eigenvalues (Enter your
answers from smallest to largest.) (λ1, λ2, λ3) = the corresponding
eigenvectors x1 = x2 = x3 =

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]

The matrix [−1320−69] has eigenvalues λ1=−1 and
λ2=−3.
Find eigenvectors corresponding to these eigenvalues. v⃗ 1= ⎡⎣⎢⎢
⎤⎦⎥⎥ and v⃗ 2= ⎡⎣⎢⎢ ⎤⎦⎥⎥
Find the solution to the linear system of differential equations
[x′1 x′2]=[−13 20−6 9][x1
x2] satisfying the initial conditions
[x1(0)x2(0)]=[6−9].
x1(t)= ______ x2(t)= _____

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

Q‒5. [8+4+8 marks] Let
Find the eigenvalues of A and the corresponding
eigenvectors.
Find a matrix P and a diagonal matrix D such
thatD=P-1AP .
Using the
equationD=P-1AP ,
computeA27 .

Suppose A is a real 2x2 matrix with complex eigenvalues α ± i β
, β ≠ 0. It was shown in class that the corresponding eigenvectors
will be complex. Suppose that a + i b is an eigenvector for α + i β
, for some real vectors a , b . Show that a − i b is an eigenvector
corresponding to α − i β . Hint: properties of the complex
conjugate may be useful. Please show...

Let A be an symmetric matrix. Assume that A has two different
eigenvalues ?1 ?= ?2. Let v1 be a ?1-eigenvector, and v2 be and
?2-eigenvector. Show that v1 ? v2. (Hint: v1T Av2 = v2T Av1.)

Normally, we start with a matrix and find the eigenvalues and
eigenvectors. But it’s interesting to see if this process can be
performed in reverse.
Suppose that a 2x2 matrix has eigenvalues of +2 and -1 but no
info on the eigenvectors. Can you find the matrix? How many
matrices would have these eigenvalues?

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