TBIL-LA Eigenvectors and Eigenspaces (GT4)

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Section 5.4 Eigenvectors and Eigenspaces (GT4)

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Learning Outcomes

Find a basis for the eigenspace of a $4 \times 4$ matrix associated with a given eigenvalue.

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Subsection 5.4.1 Class Activities

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Activity 5.4.1.

It's possible to show that $- 2$ is an eigenvalue for $[\begin{array}{ccc} - 1 & 4 & - 2 \\ 2 & - 7 & 9 \\ 3 & 0 & 4 \end{array}] .$

Compute the kernel of the transformation with standard matrix

A - (- 2) I = [\begin{array}{ccc} ? & 4 & - 2 \\ 2 & ? & 9 \\ 3 & 0 & ? \end{array}]

to find all the eigenvectors $\vec{x}$ such that $A \vec{x} = - 2 \vec{x} .$

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Definition 5.4.2.

Since the kernel of a linear map is a subspace of $R^{n},$ and the kernel obtained from $A - λ I$ contains all the eigenvectors associated with $λ,$ we call this kernel the eigenspace of $A$ associated with $λ .$

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Activity 5.4.3.

Find a basis for the eigenspace for the matrix $[\begin{array}{ccc} 0 & 0 & 3 \\ 1 & 0 & - 1 \\ 0 & 1 & 3 \end{array}]$ associated with the eigenvalue $3 .$

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Activity 5.4.4.

Find a basis for the eigenspace for the matrix $[\begin{array}{cccc} 5 & - 2 & 0 & 4 \\ 6 & - 2 & 1 & 5 \\ - 2 & 1 & 2 & - 3 \\ 4 & 5 & - 3 & 6 \end{array}]$ associated with the eigenvalue $1 .$

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Activity 5.4.5.

Find a basis for the eigenspace for the matrix $[\begin{array}{cccc} 4 & 3 & 0 & 0 \\ 3 & 3 & 0 & 0 \\ 0 & 0 & 2 & 5 \\ 0 & 0 & 0 & 2 \end{array}]$ associated with the eigenvalue $2 .$

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Subsection 5.4.2 Videos

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Figure 68. Video: Finding eigenvectors

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Exploration 5.4.6.

Given a matrix

A,

let

{\vec{v_{1}}, \vec{v_{2}}, \dots, \vec{v_{n}}}

be the eigenvectors with associated distinct eigenvalues

{λ_{1}, λ_{2}, \dots, λ_{n}} .

Prove the set of eigenvectors is linearly independent.

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Subsection 5.4.6 Sample Problem and Solution

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Sample problem Example B.1.24.

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