Matrix Algebra

A matrix is a rectangular array with rows and columns. The entry in the th row and th column is denoted by .

Which is abbreviated as when the size is known. The set of all matrices with real entries is denoted by .

Two matrices are equal if for all

Let . If for all , then

See vectors

A matrix is square is . That is, it has the same number of rows and columns.

The matrix with zero entries is called the zero matrix, denoted by , or sometimes just if the size is clear.

A matrix is symmetric if . That is, if its transpose is the same as itself.

To be symmetrical, a matrix must be a square matrix.

Let and . The matrx product is the matrix

In general,

(See transpose, matrix-vector product)

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.

Let
This means
Then

We also observe that:

Non-commutative

1. Identity matrix
2. Associative
3. Distributive (order matters)

• Associative with scalar:
• Transpose , which works for any number of variables

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.