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Use the inverse power method to estimate the eigenvector corresponding to the eigenvalue with smallest absolute value for the matrix
-1 -2 -1
A= -2 -4 -3
2 2 1
where X0= [1,1,-1].
In finding A-1 use exact arithmetic with fractions.
ln applying the power method, calculate each of the next three iterates after Xo correct to 4 significant digits.

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Solution Summary

The inverse power method is used to estimate an eigenvector. the solution is detailed and well presented.

For an n x n matrix A, show that if one or more of the eigenvalues is zero, A has no inverse.
Also show that if, A does have an inverse, the eigenvalues of A^-1 are the reciprocals of the eigenvalue A.

I have difficulties understanding an easy method to find the generalized eigenvectors of a nilpotent matrix.
For example, for the matrix A= 1st : 1 0 0
2nd; -1 2 0
3rd: 1 1 2
The first eigenvalue is 1, and the

D and E are nxn matrices, E is invertible, DE = ED, and u is an eigenvector for D corresponding to x=5.
a. Show that Eu is also an eigenvector for D corresponding to x=5.
b. Show that u is an eigenvector for D^2.
c. Show that u is an eigenvector for
D^2 - 3D.

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Show that v is an eigenvector of A and find the corresponding eigenvalue.
Show that lamda is an eigenvalue of A and find one eigenvector corresponding to this eigenvalue.
8. A = [ 2 2, 2 -1], lamda = -2
10. A = [0 4, -1 5]; lamda = 4
Use the method of Example 4.5 to find

Determining the equation for a matrix and confirming the inverse, eigenvalues
Details:
I have think the answer to question (a) is theta^2 [(1-p)I + pJ]. But when I am multiplying the sigma and the sigma inverse I still have "stuff" at the tail end of the identity......being added. I know I should be left with the identity if t

Suppose there exists a linear operator A that has an eigenvector |psi> with eigenvalue a,
A|psi> = a|psi>
Suppose that there also exists a linear operator B such that
[A, B] = B + 2BA^2
Show that the vector B|psi> is an eigenvector of A and find the associated eigenvalue.

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Given the following matrices A, B, and C compute the eigenvalue and eigenvector for each matrix.
A =(-2, 6, 6, 3)
B= (4, 2, -1, 6)
C= (0, -1, 1, 2)

Find the Inverse LaPlace Transform using different methods described in the attachment.
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Find the inverse of a matrix.
Matrix row 1 = [-1 2] row 2 = [1 3]
Find the inverse of this matrix.
a) Find A-1
b) Find A3
c) Find (A-1)3
d) Use your answers to (b) and (c) to show that (A-1)3 is the inverse of A3.