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Matrices Eigenvalues, eigenspace, and eigenbasis.

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Matrices Eigenvalues, eigenspace, and eigenbasis.

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For each of the matrices, find all (real) eigenvalues. Then find a basis of each eigenspace, and find an eigenbasis, if you can.
[■(7&8@0&9)]
[■(1&1@1&1)]
[■(6&3@2&7)]
[■(0&-1@1&2)]
[■(1&1&0@0&2&2@0&0&3)]
[■(1&1&0@0&1&1@0&0&1)]
[■(■(0&0@0&1)&■(0&0@0&1)@■(0&0@0&0)&■(0&0@0&1))]

Solution Summary

Matrices eigenvalues, eigenspaces and eigenbasis is examined.

Solution Preview

1.
The matrix is:
(1.1)
This is an upper triangular matrix, hence the eigenvalues are the elements of the diagonal:
(1.2)
The first eigenvector:

(1.3)
The second eigenvector:

(1.4)
We have two distinct eigenvalues and two distinct eigenvectors. The algebraic multiplicity (m=1) of each of the eigenvalues equal their respective geometric multiplicity, hence these two eigenvectors form an eigenbasis.

2.
The matrix is:
(2.1)
The eigenvalues are:

(2.2)
The first eigenvector:

Therefore:
(2.3)
The second eigenvector:

(2.4)
As before we have two distinct eigenvalues and two distinct eigenvectors. The algebraic multiplicity (m=1) of each of the eigenvalues equal their respective geometric multiplicity, hence these two eigenvectors form an eigenbasis. ...

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