# Matrices Eigenvalues, eigenspace, and eigenbasis.

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Matrices Eigenvalues, eigenspace, and eigenbasis.

See attached file.

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))]

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##### Solution Summary

Matrices eigenvalues, eigenspaces and eigenbasis is examined.

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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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