Matrix Algebra

The Basics of Matrix Algebra

If A is an m x n matrix and B is an n x p matrix, AB will result in an m x p matrix.

$$Given\ that\ the\ columns\ of\ b\ are\ b_1,b_2,b_p\\ AB= [Ab_1, Ab_2, Ab_3, Ab_p]$$

Here is an example using a 2x2 A matrix and a 2x3 B matrix.

$$A = \begin{bmatrix} 2 & 3 \\ 1 & 5 \end{bmatrix} B = \begin{bmatrix} 4 & 6 & 8\\ 2 & 4 & 6 \end{bmatrix}$$

$$AB= \begin{bmatrix} \begin{bmatrix} 2 & 3 \\ 1 & 5 \end{bmatrix} \begin{bmatrix} 4\\ 2 \end{bmatrix} & \begin{bmatrix} 2 & 3 \\ 1 & 5 \end{bmatrix} \begin{bmatrix} 6\\ 4 \end{bmatrix} & \begin{bmatrix} 2 & 3 \\ 1 & 5 \end{bmatrix} \begin{bmatrix} 8\\ 6 \end{bmatrix} \end{bmatrix}$$

A few theorems of note:

Given A is a m x n matrix and B and C are matrices with sizes for which the following are defined, the following are true.

• A(BC) = (AB)C

• A(B + C) = AB + AC

• (B + C)A = BA + CA

• r(AB) = (rA)B = A(rB)

However, the following are typically not true.

• AB = BA

• AB = AC meaning B = C

• AB = 0 therefore A = 0 or B = 0

Transposing Matrices

The transpose of A, denoted as A^T is a matrix whose columns are found by the rows of A.

$$A= \begin{bmatrix} 1 & 5 & 6\\ 2 & 0 & 7 \end{bmatrix} A^T= \begin{bmatrix} 1 & 2\\ 5 & 0\\ 6 & 7 \end{bmatrix}$$

Of course, there are a few theorems associated with transposing as well.

• (A^T)^T = A

• (A + B)^T = A^T + B^T

• (rA^T) = rA^T

• (AB)^T = B^T A^T

Inverse Matrices

A square matrix is invertible if it satisfies the following conditions.

The matrix C is the inverse matrix of A. C is unique. Inverse matrices are denoted by ^-1. We can use the determinant to find out of a matrix has an inverse and what the inverse is.

The determinant is denoted as follows.

$$Given\ matrix\ A\\ A= \begin{bmatrix} a & b\\ c & d \end{bmatrix}\\ The\ determinant\ is\\ ad-bc$$

If the determinant does not equal zero, there exists an inverse.

$$A^{-1}= \frac{1}{ad-bc} \begin{bmatrix} d&-b\\ -c&a \end{bmatrix}$$

If A has an inverse, then for each b in R^n, the equation Ax = b has a unique solution, x = A^-1b.

Also note the effects of inversion on the products of invertible matrices.

$$(AB)^{-1}=B^{-1}A^{-1}$$

The order of the above theorem is important. It cannot be swapped.

Inverses of non-square matrices

The inverse of a non-square matrix can be found by row-reducing the following matrix.

$$[A\ |\ I_n]$$

Reminder that I_n is an n x n identity matrix.

By row-reducing A to get I_n, you should have this matrix.

$$[I_n\ |\ A^{-1}]$$

If A does not reduce to I_n, A is not invertible.

The Invertible Matrix Theorem

There are an absolute slew of logically equivalent statements to the invertible matrix theorem, which is simply that A is an invertible matrix.

If A^-1 exists, then the following also holds true.

• A is row equivalent to I_n.

• A has n pivot positions.

• Ax = 0 only has the trivial solution.

• Columns of A are linearly independent.

• The transformation T: x -> Ax is one-to-one.

• The equation Ax = b has at least one solution for each b in R^n (the unique solution, A^-1b).

• The columns of A must span all of R^n

• T: x-> Ax is onto R^n

• There exists a matrix D that AD = I_n

• The transpose of T is invertible.

Subspaces

A subspace of R^n is defined as a set H in R^n that satisfies the following conditions:

• Vector O is in H

• H is closed under vector addition — meaning that if U and V are in H, then U + V must be in H.

• H is closed under scalar multiplication — meaning that if u is in H than Cu must be in H.

The column space of a matrix (with the notation Col A) is the set of all linear combinations of the columns of A. Col A is a set of vectors that live in R^m.

The null space of matrix A is denoted by all vectors x such that Ax = 0. Notation is Nul A.

Nul A is the set of vectors that live in R^n.

A basis for a subspace H in R^n is a set of vectors that do 2 things.

• The set of vectors is linearly independent in the subspace H.

• Those vectors span all of the subspace H.

The pivot columns of A forms a basis for Col A. NOT the pivot columns of a row reduced version of A.