Eigen Vector

Section 5.1 Eigen Vector

Goals.

We will

consider vectors with special properties with respect to a matrix,
develop a method for finding these vectors, and
discover a property of the set of these vectors.

We have considered matrices as transformations: they map vectors to new vectors. We are looking for vectors whose direction is not changed.

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Subsection 5.1.1 Motivation

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Guido is sailing with intended velocity

\vec{x} .

The effect of the wind on him is given by

A \vec{x} .

Find directions (

\vec{x}

) Guido can sail so that the wind is either directly at his back (helping him), or directly in his face (suggesting he turn around).

A = [\begin{array}{rr} 4 & - 2 \\ - 3 & 9 \end{array}] .

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We want to solve the following equation.

A \vec{x} = λ \vec{x}

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Because the vector is the same on both sides and

λ

is just a scalar, this equation says that the matrix does not change the direction of the vector. Note in this illustration

λ

indicates the magnitude of help; it is positive for a tailwind or negative for a headwind.

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Solutions for

A

are

[- 1, 3]^{T}

and

[2, 1]^{T} .

Note

[\begin{array}{rr} 4 & - 2 \\ - 3 & 9 \end{array}] [\begin{array}{r} - 1 \\ 3 \end{array}] = [\begin{array}{r} - 10 \\ 30 \end{array}] = 10 [\begin{array}{r} - 1 \\ 3 \end{array}] .

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Also

[\begin{array}{rr} 4 & - 2 \\ - 3 & 9 \end{array}] [\begin{array}{r} 2 \\ 1 \end{array}] = [\begin{array}{r} 6 \\ 3 \end{array}] = 3 [\begin{array}{r} 2 \\ 1 \end{array}]

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Note because scale does not matter if

\vec{x}

is a solution then so is

c \vec{x} .

T

is the linear transformation with matrix

A

these are vectors that are mapped onto the same line. View the results in the following images. For example

[6, 3]^{T}

also works.

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If we apply this transformation to an image, points on these two vectors will be stretched but not rotated while the rest may also rotate.

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Figure 5.1.1. Illustration of limited effect of transformation on its eigenvectors

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Definition 5.1.2. Eigenvectors and Eigenvalues.

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A vector

\vec{x}

is an \textit{eigenvector} of a matrix

M

if and only if

A \vec{x} = λ \vec{x}

for some number

λ

called the corresponding eigenvalue.

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Subsection 5.1.2 Method

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First note the following, algebraic re-arrangement of the eigenvector definition.

\begin{aligned} A \vec{x} = & λ \vec{x} . \\ A \vec{x} - λ \vec{x} = & \vec{0} . \\ (A - λ I) \vec{x} = & \vec{0} . \end{aligned}

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Activity 5.1.1.

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Remember the eigenvectors of

A

are the solutions to

(A - λ I) \vec{x} = \vec{0} .

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(a)

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The set of all such

\vec{x}

is what with respect to the matrix

A - λ I ?

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(b)

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Given we want the non-trivial solutions to

(A - λ I) \vec{x} = \vec{0}

what are we supposing is true of

A - λ I ?

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(c)

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Using the previous conclusion what do we know about

det (A - λ I) ?

Checkpoint 5.1.3.

Find the eigenvalues and eigenvectors for

B = [\begin{array}{rr} 5 & 0 \\ 2 & 1 \end{array}] .

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Definition 5.1.4. Characteristic Equation.

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The polynomial in

λ,

det (A - λ I),

is called the characteristic equation of

A .

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Subsection 5.1.3 Independence of Eigenvectors

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Next we consider the set of eigenvectors and ask if it will be independent.

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Activity 5.1.2.

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Consider eigenvectors

{\vec{v}}_{1}, {\vec{v}}_{2}, \dots, {\vec{v}}_{n}

that correspond to distinct eigenvalues

λ_{1}, λ_{2}, \dots, λ_{n} .

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Suppose

{\vec{v}}_{p + 1}

is dependent on the independent eigenvectors

{\vec{v}}_{1}, \dots, {\vec{v}}_{p} .

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(a)

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Write

c_{1} {\vec{v}}_{1} + c_{2} {\vec{v}}_{2} + \dots + c_{n - 1} {\vec{v}}_{p} = {\vec{v}}_{p + 1} .

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(b)

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Multiply this equation by

A

and simplify remembering the definition of eigenvectors.

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(c)

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Again write

c_{1} {\vec{v}}_{1} + c_{2} {\vec{v}}_{2} + \dots + c_{n - 1} {\vec{v}}_{p} = {\vec{v}}_{p + 1} .

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(d)

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Multiply this copy by

λ_{p + 1} .

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(e)

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Subtract the two equations you generated.

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(f)

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Remembering that

{\vec{v}}_{1}, \dots, {\vec{v}}_{p}

are independent, what does the resulting equation imply?

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Theorem 5.1.5.

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If the eigenvectors correspond to distinct eigenvalues, the eigenvectors are independent.

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