Matrix-Matrix Multiplication

Definition

Matrix-matrix multiplication combines two matrices.

If matrices represent transformations, multiplying them creates a new transformation.

Basic form:

A * B = C

C is the matrix that represents applying B and then A.

Matrix-matrix multiplication combines transformations.

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Shape Requirement

For:

A * B

the number of columns in A must match the number of rows in B.

Example:

A is 2 x 3
B is 3 x 4

Then:

A * B is valid

The result shape is:

2 x 4

Inner dimensions must match. Outer dimensions become the result shape.

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Why The Shape Rule Exists

A matrix transformation needs an input of the right size.

Example:

B maps 4-component vectors to 3-component vectors
A maps 3-component vectors to 2-component vectors

Then applying B first gives something A can accept:

4 components -> B -> 3 components -> A -> 2 components

So:

A * B maps 4 components to 2 components

Matrix multiplication works when the output of one transformation fits the input of the next.

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Entry Calculation

Each entry in the result comes from a row-column dot product.

Example:

A =
[1 2]
[3 4]

B =
[5 6]
[7 8]

Result entry at row 1, column 1:

dot([1, 2], [5, 7]) = 1*5 + 2*7 = 19

Result entry at row 1, column 2:

dot([1, 2], [6, 8]) = 1*6 + 2*8 = 22

Matrix multiplication uses row-column alignment.

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Full Example

Using:

A =
[1 2]
[3 4]

B =
[5 6]
[7 8]

The result is:

A * B =
[19 22]
[43 50]

Because:

row 1 with column 1 = 19
row 1 with column 2 = 22
row 2 with column 1 = 43
row 2 with column 2 = 50

Every result entry comes from one row of the first matrix and one column of the second.

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Order Matters

Matrix multiplication usually depends on order.

In general:

A * B is not the same as B * A

Why?

Because transformations happen in sequence.

Changing the order changes the sequence.

Matrix multiplication is usually not order-swappable.

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Why It Matters

Matrix-matrix multiplication lets us combine multiple transformations into one.

Example:

first scale
then rotate
then project

These can be represented by separate matrices.

Multiplying them can produce one matrix that performs the combined transformation.

Matrix multiplication composes structured vector transformations.

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Final Summary

• Matrix-matrix multiplication combines matrices.
• It is valid when inner dimensions match.
• The result shape comes from the outer dimensions.
• Each result entry is a row-column dot product.
• Order usually matters.
• Multiplication can combine transformations into one transformation.