Eigenvalue

Definition

An eigenvalue is the scale factor applied to an eigenvector.

If:

A * v = lambda * v

then:

v = eigenvector
lambda = eigenvalue

lambda is just a scalar.

Eigenvalue = how much an eigenvector is scaled.

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Simple Example

Let:

A =
[2 0]
[0 3]

Apply it to:

v = [1, 0]

Result:

A * v = [2, 0]

This is:

2 * [1, 0]

So:

eigenvector = [1, 0]
eigenvalue = 2

The eigenvalue tells the scaling amount for that eigenvector.

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Another Example

Using the same matrix:

A =
[2 0]
[0 3]

Apply it to:

v = [0, 1]

Result:

A * v = [0, 3]

This is:

3 * [0, 1]

So:

eigenvalue = 3

Different eigenvector directions can have different eigenvalues.

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Eigenvalue Meaning

An eigenvalue describes what happens along an eigenvector direction.

If:

eigenvalue = 3

the direction is stretched by 3.

If:

eigenvalue = 1

the direction is unchanged in size.

If:

eigenvalue = 0

the direction is collapsed to zero.

If:

eigenvalue = -1

the direction is flipped.

Eigenvalues describe scaling, collapse, or flip along special directions.

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Eigenvalue Depends On Eigenvector

Eigenvalues belong to eigenvector directions.

Example:

[1, 0] has eigenvalue 2
[0, 1] has eigenvalue 3

for:

[2 0]
[0 3]

The matrix does not have just one scaling behavior.

It can scale different directions differently.

Eigenvalues describe direction-specific scaling.

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Not A General Scale For Every Vector

An eigenvalue does not necessarily describe what happens to every vector.

Example:

v = [1, 1]

under:

[2 0]
[0 3]

becomes:

[2, 3]

This is not a single scalar multiple of [1, 1].

So the eigenvalue idea applies to eigenvectors.

Eigenvalues apply to preserved directions, not arbitrary vectors.

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Why Eigenvalues Matter

Eigenvalues help describe the internal behavior of a matrix transformation.

They answer:

Which stable directions are stretched?
Which are shrunk?
Which are flipped?
Which are collapsed?

Eigenvalues explain what happens along eigenvector directions.

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Final Summary

• An eigenvalue is a scalar scale factor.
• It belongs to an eigenvector direction.
• It tells how much that direction is stretched, shrunk, flipped, or collapsed.
• Different eigenvectors can have different eigenvalues.
• Eigenvalues do not describe every vector directly.
• They reveal direction-specific behavior of a matrix.