Matrix language

A matrix is a rectangular array of values set out in rows and columns. We write

:

An element is a single value in a specified matrix position. A row runs horizontally; a column runs vertically.

The order of a matrix gives the number of rows first, followed by the number of columns. Thus, an $m\times n$ matrix has $m$ rows and $n$ columns, where $m$ and $n$ are positive integers with no units. Read the order as “rows by columns”. Entries may be dimensionless, or they may have the units of the quantities they represent.

A row matrix has one row, whereas a column matrix has one column. When the number of rows equals the number of columns, the matrix is a square matrix.

Equality, addition, subtraction and scalar multiplication

Two matrices are equal if they have the same order and their corresponding elements are equal. Matching totals or determinants isn’t sufficient. Every entry must agree with the entry in the same position.

Addition and subtraction are possible only for matrices of the same order.

These operations work element by element:

A scalar is one number used to multiply every element in a matrix.

Technology can carry out each of these operations, but check the matrix order before entering a calculation. An error message often indicates that the operation is not dimensionally compatible; it doesn’t necessarily mean the calculator has failed.

Matrix multiplication

Matrix multiplication represents a sequence of linear combinations. Suppose $A$ has order $m\times n$ and $B$ has order $n\times p$, where $p$ is the positive integer giving the number of columns of $B$. The product $AB$ then exists and has order $m\times p$. The inner dimensions must match, and the outer dimensions determine the order of the product.

If $C=AB$, where $C$ is the product matrix, then

In practical terms, choose a row from the first matrix and a column from the second. Multiply corresponding elements, then add those products.

Matrix multiplication is useful when several categories interact. For instance, multiplying a matrix of product quantities by a matrix of prices or unit profits allows revenue or profit to be compared across several weeks. Units offer a quick check: quantities multiplied by price per item give money.

Properties of multiplication

For matrices with compatible orders, multiplication is associative:

Multiplication is also distributive over addition:

Matrix multiplication is generally non-commutative, however, so reversing the order may change the result:

In some cases, only one product exists. In others, both exist but differ, though they may occasionally be equal. Never reverse the order without justification. In applications, that order also records which process takes place first.

Identity and zero matrices

An identity matrix is square, with ones along its main diagonal and zeros everywhere else. We denote it by $I_n$

A zero matrix has zero in every position. It is written as $0$, or as $0_{m\times n}$ when the order must be shown explicitly

It acts as the additive identity:

Don’t confuse the number zero with a zero matrix. Its intended order must be clear from the context.

Determinants and inverses

The determinant is a scalar calculated from a square matrix. Among other things, it shows whether that matrix is invertible. For

An inverse matrix is a square matrix that gives the identity matrix when multiplied by the original matrix. The inverse of $A$ is denoted by $A^{-1}$ and satisfies

For a $2\times 2$ matrix,

provided $ad-bc\ne0$. When the determinant is zero, no inverse exists. Such a matrix is singular, meaning that it is a square matrix with no multiplicative inverse. The inverse method can be remembered as “swap the diagonal entries, change the signs of the other two, then divide by the determinant”. Calculate the determinant first.

For larger $n\times n$ matrices, use technology to find determinants and inverses. Enter the matrix carefully and calculate its determinant to check that it is invertible before using the inverse command. With $2\times 2$ matrices, be ready to do the calculation either by hand or with technology.

Systems in matrix form

A system of linear equations can be written compactly as

The units of the components depend on the model. Order is significant: each row represents one equation, while each column of $A$ corresponds to one unknown.

If $A$ is invertible, multiply on the left by $A^{-1}$:

This approach can model mixtures, production plans, prices or other situations with simultaneous constraints. In the assessed inverse-matrix setting, the coefficient matrix is taken to be invertible, except when a different condition is deliberately imposed while finding eigenvectors.

Coding and decoding

A numerical message can be placed in a column matrix, then encoded by multiplying it by an invertible matrix. If

To decode the message, reverse the operation:

The block size must match the order of the coding matrix, and the multiplication order cannot be changed. This gives a clear example of why invertibility matters. A singular coding matrix loses information, so the original message cannot always be recovered uniquely.

Matrices also link naturally to transition matrices and Markov chains, where multiplication updates a state. In phase portraits, matrix-based systems describe change in several variables. Their ability to represent higher-dimensional relationships raises a wider question: do these successful connections reveal an underlying mathematical structure in reality, or do they result from the representations humans choose? Algebra can also describe spaces beyond direct perception, but a consistent solution in a higher dimension does not by itself establish that such a physical space exists.

Eigenvalues and eigenvectors

An eigenvector is a non-zero column vector that keeps the same direction when multiplied by a square matrix. The scalar factor that scales the matching eigenvector is its eigenvalue. Let $M$ be a square matrix representing a transformation, with entries dimensionless unless the model assigns units. Its defining equation is

In a purely numerical model, both are dimensionless.

The vector might be stretched, compressed or reversed. If $\lambda>1$, its magnitude grows. If $0<\lambda<1$, it shrinks; if $\lambda<0$, it is scaled and its direction reverses. Multiplying an eigenvector by any non-zero scalar produces another eigenvector with the same eigenvalue, so eigenvectors aren't normally unique.

For a known eigenvalue, find an eigenvector by rearranging the defining equation:

This homogeneous system must have a non-zero solution. Therefore, $M-\lambda I_2$ has to be singular.

Characteristic polynomial

The characteristic polynomial is the polynomial whose roots give the eigenvalues of a square matrix. For a $2\times2$ matrix,

The eigenvalues satisfy

For

this becomes

Solve the quadratic to get the eigenvalues. Then substitute each one into $(M-\lambda I_2)\mathbf{v}=\mathbf{0}$ and find a corresponding eigenvector. Only $2\times2$ matrices require hand calculations and technology here. When using technology, retain enough precision to tell genuinely equal values apart from those that only appear close after rounding.

Diagonalization

A matrix is diagonalizable if it can be written using an invertible matrix of eigenvectors and a diagonal matrix containing the corresponding eigenvalues. The required case here involves two distinct real eigenvalues.

Let $\lambda_1$ and $\lambda_2$ be the distinct real eigenvalues of $M$. Let $\mathbf{v}_1$ and $\mathbf{v}_2$ be their corresponding non-zero eigenvectors, with the subscripts showing the matched eigenvalue–eigenvector pairs. Define

Then

The order matters. If the columns of $P$ are swapped, the diagonal entries of $D$ must be swapped too. Distinct real eigenvalues produce linearly independent eigenvectors, making $P$ invertible.

Diagonalization can also be viewed as a change of coordinates. First, $P^{-1}$ converts a vector into eigenvector coordinates. Next, $D$ scales those coordinates independently, before $P$ converts the result back.

Powers of a matrix

Repeated matrix multiplication quickly becomes cumbersome, whereas powers of a diagonal matrix are immediate. If $t$ is a non-negative integer representing the number of repeated steps, then

with

The formula works because each intermediate pair $P^{-1}P$ collapses to the identity. It also reveals long-term behaviour: the eigenvalue with the greatest absolute value usually determines the dominant direction, provided the initial state has a component in its eigenvector direction.

Discrete models

Suppose $\mathbf{s}_t$ is a state column at step $t$, and its components carry the units of the quantities being modelled. The state one step later is $\mathbf{s}_{t+1}$. For a two-category model, the recurrence is

If $\mathbf{s}_0$ is the initial state column, then

This structure can model a population moving between two locations, predator and prey population interactions, stochastic transitions, or changes in two related financial values. Interpret the components and their units after calculating. A negative population usually shows that the model or its assumptions are unsuitable; it isn't a meaningful prediction.

An invariant state is a state vector left unchanged by the transformation. If $\mathbf{s}_*$ is an invariant state, with the star labelling the unchanged state, then

So an invariant state is an eigenvector associated with eigenvalue $1$. This idea plays a central role in the long-run behaviour of stochastic processes and ranking methods.

The same matrix structure occurs in coupled differential equations and their phase portraits. Continuous models, however, describe change at every instant rather than at separate steps. Matrices can also represent transformations of conics and help analyse linked market values and trends.

Why eigenvalues matter

Eigenvalues reveal the characteristic modes of a system. Engineers use them to analyse vibration and damping, detect structural defects, interpret subsurface measurements and avoid destructive resonance. Large-scale ranking algorithms rely on dominant eigenvectors to assign relative importance within a network. There is an ethical dimension to these uses: a model may influence safety decisions or the visibility of information, so its assumptions and limitations matter.

In data analysis, principal component methods use eigenvectors to find important directions of variation. Factor methods instead seek a smaller collection of underlying influences. These are useful extensions, not additional calculation requirements here.

Eigenvalues also link discrete updates to continuous dynamical systems. Repeated multiplication gives powers such as $\lambda^t$, while continuous growth naturally uses Euler’s number $e$, where $e$ is the dimensionless base of the natural logarithm. This close fit between matrix models and observed processes raises a broader question: was mathematics designed to describe the world, or does its success suggest that the world itself has an intrinsically mathematical structure?