Identity Matrix

Definition

The identity matrix is a matrix that leaves vectors unchanged.

Example in two dimensions:

I =
[1 0]
[0 1]

Apply it to:

v = [3, 5]

Result:

I * v = [3, 5]

Identity matrix = do-nothing transformation.

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Why It Leaves Vectors Unchanged

Use:

I =
[1 0]
[0 1]

and:

v = [3, 5]

Compute:

first output  = 1*3 + 0*5 = 3
second output = 0*3 + 1*5 = 5

Output:

[3, 5]

The identity matrix preserves each component.

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Identity Matrix Shape

Identity matrices are square.

Examples:

1 x 1:
[1]

2 x 2:
[1 0]
[0 1]

3 x 3:
[1 0 0]
[0 1 0]
[0 0 1]

They have 1s on the main diagonal and 0s elsewhere.

An identity matrix matches the dimension of the vectors it preserves.

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Identity Matrix And Multiplication

The identity matrix acts like 1 for matrix multiplication.

Example:

I * A = A
A * I = A

when the shapes match.

This means applying identity before or after a transformation does not change the transformation.

Identity matrix is the neutral element for matrix multiplication.

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Identity vs Zero Matrix

The identity matrix is not the zero matrix.

Identity:

[1 0]
[0 1]

Zero matrix:

[0 0]
[0 0]

Identity preserves vectors.

Zero matrix sends every vector to zero.

Example:

zero matrix * [3, 5] = [0, 0]

Identity preserves information. Zero matrix removes component values.

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Why It Matters

The identity matrix gives a reference point for transformations.

It answers:

What does no transformation look like?

This matters when studying inverse matrices.

An inverse should undo a transformation and return us to identity behavior.

Identity is the baseline transformation.

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Final Summary

• The identity matrix leaves vectors unchanged.
• It has 1s on the main diagonal and 0s elsewhere.
• It is square.
• It acts like 1 in matrix multiplication.
• It is different from the zero matrix.
• It is the baseline for understanding inverse transformations.