Matrix Basics

Linear Algebra deals primarily with matrices, which are boxes of numbers. Here's an example of a matrix:

$$ \large A = \begin{bmatrix} 3 & -1 & 4\\ -5 & 0 & 2\\ \end{bmatrix} $$

We typically use capital letters to denote matrices. The matrix $\large A$ in the example above is a $\large 2\times 3$ matrix, which means it has 2 rows and 3 columns. To pick out an individual element of a matrix (one number in the box), we use a subscript notation. In the example above, we could write $\large A_{1,2}=-1$ because the element in the first row and second column is a $\large -1$.

Important

By convention, we always use a (row,column) convention when working with matrices.

Note

Most matrices we encounter in STEM applications are very large, with hundreds or thousands of rows and/or columns. In this class, we'll do all our hand calculations using small matrices with up to 5-6 rows/columns. We'll work with larger matrices in our Python labs.

Basic Definitions and Calculations

This is a list of some basic matrix operations with examples:

• Addition/subtraction works by corresponding components:

$$\large \begin{bmatrix} 2 & 3 \\ 4 & -1 \\ \end{bmatrix} + \begin{bmatrix} 1 & 0 \\ -2 & 6 \\ \end{bmatrix} = \begin{bmatrix} 3 & 3\\ 2 & 5 \\ \end{bmatrix}$$

• Scalar multiplication also works by component:

$$\large 3\begin{bmatrix} 6 & -4 \\ 2 & 0 \\ 3 & 1 \\ \end{bmatrix} = \begin{bmatrix} 18 & -12\\ 6 & 0 \\ 9 & 3\\ \end{bmatrix}$$

• The transpose of a matrix, denoted $\large A^T$, is the matrix we get if we turn the rows of $\large A$ into columnns (or, equivalently, the columns into rows):

$$\large \begin{bmatrix} 6 & -4 \\ 2 & 0 \\ 3 & 1 \\ \end{bmatrix}^T = \begin{bmatrix} 6 & 2 & 3\\ -4 & 0 & 1 \\ \end{bmatrix}$$

• A square matrix is a matrix that has the same number of rows and columns (a $\large 2\times 2$, a $\large 3\times 3$, etc.)

• The identity matrix $\large I$ is a square matrix with this form:

$$\large I = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \\ \end{bmatrix} $$

• A zero matrix can be any size and consists only of 0's.

• Vectors can be considered 1-dimensional matrices. So,

$$\large \vec{x} = \begin{bmatrix} 3 \\ -2 \\ 1\\ \end{bmatrix} \text{, a column vector is a } 3\times 1 \text{ matrix } $$

$$\large \vec{y} = \begin{bmatrix} 4 & 1 & -1 & 5\\ \end{bmatrix} \text{, a row vector is a } 1\times 4 \text{ matrix } $$

• The diagonal of a square matrix is the line of elements connecting the upper left element to the bottom right element.

Determinants

Every square matrix is associated with a number called a determinant. For larger matrices, computing a determinant can be difficult. But we will only need to compute determinants for $\large 2\times 2$ matrices, and that computation is straightforward:

$$\large \det\begin{bmatrix} 3 & 8 \\ -1 & 6\\ \end{bmatrix} = (3)(6)-(-1)(8)=26 $$

The determinant is the difference of the product of the diagonal elements times the off-diagonal elements of the matrix.

Matrix Multiplication

The product of two matrices has a trickier definition than addition or subtraction. The product $\large C=AB$ is defined as follows:

$$\large C_{i,j}=\text{ the dot product of the ith row of A with the jth column of B}=\sum_{k=1}^nA_{i,k}B_{k,j}$$

Here's a video example of how this works:

Because of this definition, matrix multiplication is not always defined. For $\large C=AB$ to exist, the number of columns of $\large A$ must equal the number of rows of $\large B$. The following image shows the common way to see if $\large AB$ is defined, which also gives us a way to determine the size of $\large AB$. We line up the sizes of $\large A$ and $\large B$, and the inner dimensions must be equal for the the product to be defined. If the product is defined, the outer dimensions give us the size of the product.

Warning

Matrix multiplication is not commutative! That is, even if they both exist (which they often don't), $\large AB$ is not necessarily equal to $\large BA$.

Note

We will see soon why matrix multiplication has this funny definition. It's related to the way we represent systems of equations in linear algebra.

Matrix-Vector Multiplication

Since a vector is just a one dimensional matrix, matrix-vector multiplication is just a small case of matrix-matrix multiplication. For example:

$$\large \begin{bmatrix} 3 & 8 & 0\\ -1 & 6 & -2\\ \end{bmatrix} \begin{bmatrix} 4 \\ -3 \\ 1 \\ \end{bmatrix} = \begin{bmatrix} (3)(4)+(8)(-3)+(0)(1) \\ (-1)(4)+(6)(-3)+(-2)(1) \\ \end{bmatrix} = \begin{bmatrix} -12 \\ -24 \\ \end{bmatrix} $$

Multiplication as Linear Combinations (another way to see it...)

Special matrices (symmetric, orthogonal, diagonal, triangular, etc)

Section Outcomes

I can:

• Use linear algebra terminology and notation.
• Perform basic matrix computations.
• Multiply matrices together (or recognize when the matrix product does not exist).