Practical Uses of Matrix Mathematics

December 17, 2013 by Mike DeHaan 56 Comments

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What are the practical use of matrices in day to day life? Image by Lakeworks.

My thanks to an alert reader for asking, “What are the practical use of matrices in day to day life?” The most direct answer is, “It depends on your own day to day life.” Let’s consider some practical uses of matrix mathematics in a variety of settings, along with a brief introduction to matrices.

Applications of Matrix Mathematics

Matrix mathematics applies to several branches of science, as well as different mathematical disciplines. Let’s start with computer graphics, then touch on science, and return to mathematics.

We see the results of matrix mathematics in every computer-generated image that has a reflection, or distortion effects such as light passing through rippling water.

Before computer graphics, the science of optics used matrix mathematics to account for reflection and for refraction.

Matrix arithmetic helps us calculate the electrical properties of a circuit, with voltage, amperage, resistance, etc.

In mathematics, one application of matrix notation supports graph theory. In an adjacency matrix, the integer values of each element indicates how many connections a particular node has.

The field of probability and statistics may use matrix representations. A probability vector lists the probabilities of different outcomes of one trial. A stochastic matrix is a square matrix whose rows are probability vectors. Computers run Markov simulations based on stochastic matrices in order to model events ranging from gambling through weather forecasting to quantum mechanics.

Matrix mathematics simplifies linear algebra, at least in providing a more compact way to deal with groups of equations in linear algebra.

Introduction to Matrix Arithmetic

A matrix organizes a group of numbers, or variables, with specific rules of arithmetic. It is represented as a rectangular group of rows and columns, such as $begin{bmatrix} 11 & 12 & 13\ 21 & 22 & 23 end{bmatrix}$ . This “2X3” matrix has two rows and three columns; the number ’23’ is in the second row of the third column.

An example of a square matrix with variables, rather than numbers, is $begin{bmatrix} a & b\ c & d end{bmatrix}$ . This is a square matrix because the number of rows equals the number of columns.

We can only add matrices of the same dimensions, because we add the corresponding elements. $begin{bmatrix} a & b\ c & d end{bmatrix} + begin{bmatrix} e & f\ g & h end{bmatrix} = begin{bmatrix} a+e & b+f\ c+g & d+h end{bmatrix}$ .

Matrix multiplication is another matter entirely. Let’s multiply matrices MP=R. M is an mXn matrix; P is nXp; and the result R will have dimension mXp. Note that the number of columns of the left-hand matrix, M, must equal the number of rows of the right hand matrix, P. For example:

$begin{bmatrix} a & b & c\ d & e & f end{bmatrix} * begin{bmatrix} g & h\ i & j \k & l end{bmatrix} = begin{bmatrix} a*g + b*i + c*k & a*h + b*j + c*l\ d*g + e*i + f*k & d*h + e*j + f*l end{bmatrix}$ .

A matrix can also multiply, or be multiplied by, a vector.

Surprisingly, we all use matrix in our daily lives. Image by Kkmann.

Graphic Uses of Matrix Mathematics

Graphic software uses matrix mathematics to process linear transformations to render images. A square matrix, one with exactly as many rows as columns, can represent a linear transformation of a geometric object. For example, in the Cartesian X-Y plane, the matrix $begin{matrix} 0 & -1 \ 1 & 0 end{matrix}$ reflects an object in the vertical Y axis. In a video game, this would render the upside-down mirror image of a castle reflected in a lake.

If the video game has curved reflecting surfaces, such as a shiny silver goblet, the linear transformation matrix would be more complicated, to stretch or shrink the reflection.

The Identity Matrix and the Inverse Matrix

The Identity matrix is an nXn square matrix with ones on the diagonal and zeroes elsewhere. It causes absolutely no change as a linear transformation; much like multiplying an ordinary number by one. The dimension of an Identity matrix is shown by a subscript, so I₂ = $begin{matrix} 1 & 0 \ 0 & 1 end{matrix}$ is the 2X2 Identity matrix.

Suppose we have two square nXn matrices, A and B, such that AB=I_n. Then we call B the inverse matrix of A, and show it as A^-1. The first practical point is that the inverse matrix A^-1 reverses the changes made by the original linear transformation matrix A.

The Determinant

Another important task in matrix arithmetic is to calculate the determinant of a 2X2 square matrix. For matrix M= $begin{bmatrix} a & b\ c & d end{bmatrix}$ , the determinant is |M| = a*d – b*c.

If the determinant of M is zero, then no inverse matrix M^-1 exists.

On the other hand, if we apply M as the linear transformation of a unit square U into U_M, then the determinant |M| is the area of that transformed square. In a sense, the determinant is the size, or “norm”, of a square matrix.

Daily Matrix Applications

Matrix mathematics has many applications. Mathematicians, scientists and engineers represent groups of equations as matrices; then they have a systematic way of doing the math. Computers have embedded matrix arithmetic in graphic processing algorithms, especially to render reflection and refraction. Some properties of matrix mathematics are important in math theory.

However, few of us are likely to consciously apply matrix mathematics in our day to day lives.

Readers, please leave a comment: how do you use matrices on a daily basis?

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