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# Finding eigen values and eigen space

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Let A = . Find each of the following (by hand, showing your work).

(a) The characteristic polynomial of A
(b) The eigenvalues of A.
(c) The algebraic multiplicity for each eigenvalue.
(d) A basis for each eigenspace.
(e) Is "A" diagonalizable? Why or why not?

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HI,

Please find the explanations/solution attached herewith.

Let A = . Find each of the following (by hand, showing your work).
(a) The characteristic polynomial of A
(b) The eigenvalues of A.
(c) The algebraic multiplicity for each eigenvalue.
(d) A basis for each eigenspace.
(e) Is "A" diagonalizable? Why or why not?

Solution:
(a)

The characteristic equation is

(b)

Eigen values are 3 and 2.

(c)
The multiplicity of eigen value 2 is 2.
The multiplicity of eigen value 3 is 2.
(d)
The Eigen vector corresponding to =3, will be
Substitute =3 into the gives

Which gives y = z = w = 0
Therefore, the eigenvectors of A associated with the eigenvalue λ = 3 are all vectors of the form ( x, 0, 0, 0)T = x(1, 0, 0, 0)T for x≠ 0. Removing the restriction that the scalar multiple be nonzero includes the zero vector and gives the full eigenspace:

Basis for this eigenspace is .
The Eigen vector corresponding to =2, will be
Substitute =3 into the gives

Which gives z = w = 0 , x = -2y
Therefore, the eigenvectors of A associated with the eigenvalue λ = 2 are all vectors of the form ( -2y, y, 0, 0)T = y(-2, 1, 0, 0)T for y≠ 0. Removing the restriction that the scalar multiple be nonzero includes the zero vector and gives the full eigenspace:

Basis for this Eigen space is .

(e)
A is not diagonalizable as the its all eigen values are not different.

This content was COPIED from BrainMass.com - View the original, and get the already-completed solution here!

© BrainMass Inc. brainmass.com October 2, 2022, 11:47 am ad1c9bdddf>
https://brainmass.com/math/linear-algebra/finding-eigen-values-eigen-space-204062