Eigenvector

Definition

An eigenvector is a vector whose direction is preserved by a matrix transformation.

The matrix may stretch or shrink it.

But it does not rotate it away from its original direction.

Example idea:

A * v = scaled version of v

Eigenvector = a direction that stays on its own line after transformation.

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Simple Scaling Example

Let:

A =
[2 0]
[0 3]

Apply it to:

v = [1, 0]

Result:

A * v = [2, 0]

The vector changed size, but it still points along the first axis.

So [1, 0] is an eigenvector of this matrix.

If the direction stays the same, the vector may be an eigenvector.

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Another Direction

Using the same matrix:

A =
[2 0]
[0 3]

Apply it to:

v = [0, 1]

Result:

A * v = [0, 3]

The direction stays along the second axis.

So [0, 1] is also an eigenvector.

Some transformations have multiple preserved directions.

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Not Every Vector Is An Eigenvector

Using:

A =
[2 0]
[0 3]

apply it to:

v = [1, 1]

Result:

A * v = [2, 3]

[2, 3] is not just a scaled version of [1, 1].

The direction changed.

So [1, 1] is not an eigenvector of this matrix.

An eigenvector must keep its direction under the transformation.

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Eigenvectors Are About Directions

If:

v

is an eigenvector, then a nonzero scaled version of v points in the same eigenvector direction.

Example:

[1, 0]
[5, 0]
[-2, 0]

all lie on the same line.

The important thing is the preserved direction.

Eigenvectors identify special directions of a transformation.

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Zero Vector Is Excluded

The zero vector is not counted as an eigenvector.

Example:

[0, 0]

It has no direction.

Eigenvectors are about directions that are preserved.

Eigenvectors must be nonzero.

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Why Eigenvectors Matter

Eigenvectors help reveal how a matrix acts on space.

They answer:

Which directions does this transformation preserve?
Which directions are only scaled?
Which directions define the transformation's structure?

Eigenvectors expose stable directions inside a transformation.

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Final Summary

• An eigenvector is a nonzero vector whose direction is preserved.
• The matrix may stretch, shrink, or flip it.
• The result must stay on the same line.
• Not every vector is an eigenvector.
• Eigenvectors are about directions, not one specific length.
• They reveal stable directions of a matrix transformation.