Matrix Multiplication

Also see: Power of Matrix

Matrix Multiplication

Definition

Let and . The matrx product is the matrix

In general,

(See transpose, matrix-vector product)

To see what matrix multiplication represents, see Composition of Linear Transformations.

Remark

Matrix multiplication is actually the composition of two linear transformation. While this note only covers calculation, it's a good idea to go see composition of linear transformations for a conceptual understanding as to what it going on.

Example

Let
This means
Then

We also observe that:

Properties

Non-commutative

  1. Identity matrix
  2. Associative
  3. Distributive (order matters)
  • Associative with scalar:
  • Transpose , which works for any number of variables
Important

Matrix multiplication is NOT commutative. Order matters. Why? Think about geometric transformations. If we rotate counter-clockwise 90 degrees, and then reflect on the x-axis, that's a different result than reflecting on the x-axis first, then rotating 90 degrees. Visualize this.