Matrices

Matrices are more than tables of numbers—they are tools that store information and functions that transform space. Originally developed to organize and solve systems of linear equations, matrices now serve as a core language of linear algebra. When a matrix multiplies a vector, it performs a linear transformation, such as stretching, rotating, or shifting that vector into a new coordinate space. This dual nature—data container and transformation machine—makes matrices essential in many fields, from solving equations and studying eigenvalues to graph theory, representation theory, and modeling rotations. Although their notation can look intimidating, matrices are simply structured ways to organize information and powerful devices for manipulating both data and geometric space.

Contents

Definition

A matrix is a collection of numbers, called elements, arranged in a rectangle or a square. The numbers do not have to be positive, and they can be decimals or even complex numbers. A square matrix is, as the name suggests, a matrix that is square in shape, with the same number of columns and rows. In algebra, a matrix is usually represented by a capital letter in bold or underlined. The numbers in a matrix are surrounded by square or curved brackets.

Who

TBD

What

In mathematics and computer science, a matrix generally represents one of three things:

1. A System of Equations
When modeling complex systems—like economic forecasts, weather patterns, or quantum mechanics—mathematicians use systems of linear equations. A matrix acts as a compression format for these equations. Instead of writing out hundreds of separate variables over and over, you strip the variables away and arrange only the numerical coefficients into a tight grid. This allows computers to solve massive, multi-variable problems by manipulating a single block of data at once.

2. A Geometric Transformation
In 3D computer graphics, a matrix is not just static data; it is an action. It serves as a set of mathematical instructions for moving, rotating, stretching, or squashing space. When you move your mouse to rotate the camera in a video game, the computer does not manually calculate the new position of every polygon point one by one. Instead, it multiplies the entire character model by a single rotation matrix—a grid of numbers containing the exact angles for turning things. Every CGI frame you watch involves a processor crunching billions of matrix multiplications.

3. A Map of Connections (Networks)
Matrices are the ultimate tool for mapping relationships in a network, whether that means flight routes between airports, nodes on the internet, or who follows whom on social media. Mathematicians use an adjacency matrix to convert these visual webs into pure data.

A side-by-side comparison of two network graphs and the matrices used to represent their connections. Photo by Jakkritcpenu (Wikimedia Commons) is licensed under CC BY-SA 4.0.

If you look at the graphs and their matrices above, every row and column represents a node (A, B, C, etc.). If a line connects node A to node B, a 1 goes into that spot in the matrix. If there is no connection, it gets a 0. This turns a tangled spiderweb of relationships into a clean, searchable grid. Google’s original PageRank algorithm relied heavily on this concept, representing the entire World Wide Web (WWW) as one unimaginably massive matrix where every 1 and 0 represented a hyperlink from one website to another. [1]

Types of Matrices

Based on the number of rows and columns present and the special characteristics shown, types of matrices are classified into various types.

Row Matrix

A matrix that contains only one row and any number of columns is known as a row matrix or row vector.

A\,\,=\,\,\left[ \begin{matrix} 2& -3& 5& 1\\ \end{matrix} \right]

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Column Matrix

A matrix that contains only one column and any number of rows is called a column matrix.

C\,\,=\,\,\left[ \begin{array}{c} 2\\ -3\\ 5\\ 1\\ \end{array} \right]

TBD

Rectangular Matrix

A matrix that does not have an equal number of rows and columns is known as a rectangular matrix.

R\,\,=\,\,\left[ \begin{matrix} 1& 2& 3\\ 4& 5& 6\\ \end{matrix} \right]

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Square Matrix

A matrix that has an equal number of rows and an equal number of columns is called a square matrix.

S\,\,=\,\,\left[ \begin{matrix} 1& 2& 3\\ 4& 5& 6\\ 7& 8& 9\\ \end{matrix} \right]

TBD

Zero Matrix

A matrix whose all elements are zero is called a zero matrix. A zero matrix is also called a null matrix because all its elements are zero.

O\,\,=\,\,\left[ \begin{matrix} 0& 0& 0\\ 0& 0& 0\\ 0& 0& 0\\ \end{matrix} \right]

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Diagonal Matrix

A matrix that has all elements as 0 except diagonal elements is known as a diagonal matrix. A diagonal matrix is only possible in the case of a square matrix.

D\,\,=\,\,\left[ \begin{matrix} 2& 0& 0\\ 0& -3& 0\\ 0& 0& 5\\ \end{matrix} \right]

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Scalar Matrix

A diagonal matrix whose all diagonal elements are non-zero and the same is called a scalar matrix. A scalar matrix is a kind of diagonal matrix where all diagonal elements are the same. The identity matrix is a special case of scalar matrix.

B\,\,=\,\,\left[ \begin{matrix} 4& 0& 0\\ 0& 4& 0\\ 0& 0& 4\\ \end{matrix} \right]

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Identity Matrix

A diagonal matrix where all the diagonal elements are 1 and all non-diagonal elements are 0 is called an identity matrix. The identity matrix is also called the unit matrix. The identity matrix, or unit matrix, always has an equal number of rows and columns (square matrix).

I\,\,=\,\,\left[ \begin{matrix} 1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\\ \end{matrix} \right]

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Upper Triangular Matrix

A square matrix in which all the elements below the diagonal are zero and the elements from the diagonal and above are non-zero elements is called an upper triangular matrix. In an upper triangular matrix, the non-zero elements form a triangular-like shape.

U\,\,=\,\,\left[ \begin{matrix} 1& 2& 3\\ 0& 4& 5\\ 0& 0& 6\\ \end{matrix} \right]

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Lower Triangular Matrix

A square matrix in which all the elements above the diagonal are zero and the elements from the diagonal and below are non-zero elements is called a lower triangular matrix. In a lower triangular matrix, the non-zero elements form a triangular-like shape from the diagonal and below.

L\,\,=\,\,\left[ \begin{matrix} 1& 0& 0\\ 2& 3& 0\\ 4& 5& 6\\ \end{matrix} \right]

TBD

Symmetric Matrix

A square matrix “A” of any order is defined as a symmetric matrix if the transpose of the matrix is equal to the original matrix itself, i.e., A^T = A.

A\,\,=\,\,\left[ \begin{matrix} 1& 2& 3\\ 2& 4& 5\\ 3& 5& 6\\ \end{matrix} \right]

TBD

Skew-Symmetric Matrix

A square matrix “A” of any order is defined as a skew-symmetric matrix if the transpose of the matrix is equal to the negative of the original matrix itself, i.e., A^T = -A.

A\,\,=\,\,\left[ \begin{matrix} 0& -a& -b\\ a& 0& -c\\ b& c& 0\\ \end{matrix} \right]

TBD

Determinant of a Matrix

See Matrix Determinants.

Rank of a Matrix

See Matrix Rank.

Why

See Theoretical Knowledge Vs Practical Application.

TBD

How

TBS

Many of the References and Additional Reading websites and Videos will assist you with understanding and applying matrices.

As some professors say: “It is intuitively obvious to even the most casual observer.“

References

[ ℰ ] [1] OpenBlueprint. “This Might Be a Surprisingly Simple Question, but What Is a Matrix in Mathematics and What Does It Represent?” Quora, June 20, 2026. https://qr.ae/pFJe9r.

[ ℰ ] “Introduction to Matrices.” GeeksforGeeks, January 10, 2025. https://www.geeksforgeeks.org/maths/introduction-to-matrices/.

[ ℰ ] “Types of Matrices.” GeeksforGeeks, November 22, 2020. https://www.geeksforgeeks.org/maths/types-of-matrices/.

[ ℰ ] “Rank of a Matrix.” GeeksforGeeks, September 13, 2023. https://www.geeksforgeeks.org/maths/rank-of-matrix/.

Amit, Alon. “What Is an Intuitive Explanation of the Rank of a Matrix?” 2023. Quora. Accessed December 27. https://qr.ae/pKchRc.

Figueira, João Paulo. 2020. “Matrix Inversion and Meaning.” Medium. Zero Equals False. May 24. https://medium.com/zero-equals-false/matrix-inversion-and-meaning-291068cd5db1.

migmar (t9nz), tenzin. 2023. “Linear Algebra 2: Echelon Matrix Forms.” Medium. Towards Data Science. September 29. https://towardsdatascience.com/linear-algebra-2-echelon-matrix-forms-ce8a19c71a7d.

Pandey, Rohit. 2023. “A Birds-Eye View of Linear Algebra: Why Is Matrix Multiplication Like That?” Medium. Towards Data Science. November 26. https://towardsdatascience.com/a-birds-eye-view-of-linear-algebra-why-is-matrix-multiplication-like-that-a4d94067651e.

Zhang, Renda. 2023. “What Is a Matrix (Easy to Understand).” Medium. Medium. December 11. https://rendazhang.medium.com/what-is-a-matrix-easy-to-understand-f1ea14f548d8.

Additional Reading

Amit, Alon. “Are Matrix Multiplication and Normal Multiplication Really Similar?” 2023. Quora. Accessed December 27. https://qr.ae/pKch1c.

Hayrapetyan, Tigran. “Understanding Matrices.” Towards Data Science, May 26, 2025. https://towardsdatascience.com/understanding-matrices-part-1-matrix-vector-multiplication/.

In the first story of the series, which is devoted to the interpretation of algebraic matrices, we looked at how matrices can be presented geometrically, and called it “X-way interpretation”. Such interpretation explicitly highlights various properties of matrix-vector multiplication, as well as the behavior of matrices of several special types.

Hayrapetyan, Tigran. “Understanding Matrices.” Towards Data Science, June 19, 2025. https://towardsdatascience.com/understanding-matrices-part-2-matrix-matrix-multiplication/.

In the current 2nd story of this series, we observed how matrix-matrix multiplication can be presented visually, with the help of so-called “X-diagrams”. We have learned that doing multiplication “C = A*B” is the same as concatenating X-diagrams of those two matrices. This method clearly illustrates various properties of matrix multiplications, like why it is not a symmetrical operation (“A*B ≠ B*A“), as well as explains the formula:

We have also observed why multiplication behaves in certain ways when operands are matrices of special types (scale, shift, permutation, and triangular matrices).

Hayrapetyan, Tigran. “Understanding Matrices.” Towards Data Science, July 22, 2025. https://towardsdatascience.com/understanding-matrices-part-3-matrix-transpose/.

That’s all I wanted to present in regard to the matrix transpose operation. I hope that the visual methods illustrated above will help all of us to gain a better grasp of various matrix operations.

Hayrapetyan, Tigran. “Understanding Matrices.” Towards Data Science, August 30, 2025. https://towardsdatascience.com/understanding-matrices-part-4-matrix-inverse/.

In this story, we have looked at matrix inversion operation A-1 as something that undoes the transformation of the given matrix “A“. We have observed why inverting a chain of matrices like (ABC)-1 actually reverses the order of multiplication, resulting in C-1B-1A-1. Also, we got a visual perspective on why inverting several special types of matrices results in another matrix of the same type.

∑ Mathematical Mysteries. “Eigenvectors and Eigenvalues.” September 26, 2021. https://mathematicalmysteries.org/eigenvectors-and-eigenvalues/.

∑ Mathematical Mysteries. “Eigenvectors and Eigenvalues Additional Reading.” June 13, 2024. https://mathematicalmysteries.org/eigenvectors-and-eigenvalues-additional-reading/.

∑ Mathematical Mysteries. “Linear Algebra.” September 26, 2021. https://mathematicalmysteries.org/linear-algebra/.

“Matrix (Mathematics).” Wikipedia, September 14, 2026. https://en.wikipedia.org/wiki/Matrix_(mathematics).

In mathematics, a matrix (pl.: matrices) is a rectangular array of numbers or other mathematical objects with elements or entries arranged in rows and columns, usually satisfying certain properties of addition and multiplication.

Pierce, Rod. “Matrices.” Math Is Fun. Accessed September 26, 2026. https://www.mathsisfun.com/algebra/matrix-introduction.html.

Gundersen, Gregory. “A Geometrical Understanding of Matrices.” gregorygunderson.com, October 24, 2018. https://gregorygundersen.com/blog/2018/10/24/matrices/.

The goal of this post is to lay the ground work for understanding matrices as geometric objects. I focus on matrices because they effect how I think of vectors, vector spaces, the determinant, null spaces, spans, ranks, inverse matrices, singular values, and so on. In my mind, every one of these concepts is easier to understand with a strong, geometrical understanding of matrices. For simplicity, I will focus on real-valued matrices and vectors, and I error on the side of intuitive examples rather than generalizations.

AM. “A Beginner’s Guide to Matrices: Definition, Properties, Operations and Types.” Medium, March 1, 2023. https://abhishekmishra13k.medium.com/a-beginners-guide-to-matrices-definition-properties-operations-and-types-71d0978db8bb.

Videos

The Applications of Matrices | What I wish my teachers told me way earlier

The Applications of Matrices | What I wish my teachers told me way earlier

This video goes over just a few applications of matrices that may give you some insight into how they can be used in the real world. Linear algebra was never explained well to me in school and I had very little motivation to learn matrices at the time so hopefully this helps you if you’re in a similar situation. Also note there are so many applications there’s just no way to fit them into one video but here you’ll find some of my favorite instances of when they come up.

Matrices in Motion

Matrices in Motion – themathbagel

Vector

The primary difference is their dimensionality and purpose: a vector represents a single point, direction, or list of values in space, while a matrix is a two-dimensional grid of numbers that typically acts as a transformation or a system of equations.

Key Differences of a Matrix and a Vector

Feature	Vector	Matrix
Dimensions	1D (has only a single row or a single column)	2D (has rows and columns)
Notation	Written as a single column [x, y, z]^T or row [x, y, z]	Written as a grid of numbers surrounded by brackets
Size Description	Described by its length or number of elements (e.g., a “3-element vector”)	Described by rows × columns (e.g., a “3 × 4 matrix”)
Role in Math	Represents a position, velocity, or single data point	Represents a transformation (like rotation) or a system of equations

Visualizing the Concepts

Think of a vector as a single line of data. For example, the 3D coordinates of an airplane: latitude, longitude, altitude.
Think of a matrix as a spreadsheet or grid of data. For example, tracking those same 3D coordinates for an entire fleet of 50 airplanes over time.

Notes

Transpose

The superscript [x,y,z]^T means the transpose of a vector or matrix.

The transpose of a matrix is a new matrix that is obtained by swapping the rows and columns of the given matrix or vice versa, i.e., for the given matrix, the elements in rows are interchanged with the elements in columns.
The transpose of a vector changes a column vector into a row vector, or a row vector into a column vector, without changing the order of the values

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The featured image on this page is from the YouTube video Matrices.