Working With n×n Matrices in MATLAB: A Practical Walkthrough
There is no special syntax for a square matrix in MATLAB. You just create it the same way you create any other array. The confusion usually comes from people looking for something like a dedicated square-matrix constructor, and it does not exist because the language does not differentiate between square and rectangular arrays at the type level. What matters is whether your dimensions actually match for the operation you are about to run. If you are searching for a downloadable PDF that contains this material, you will find scattered examples across university course notes and documentation sites, but most of them are outdated and still reference MATLAB versions from before R2016b. The behavior of symbolic matrix operations changed noticeably around that release. I would recommend generating your own PDF using MATLAB's Report Generator or simply printing a script window to PDF. The code below is self-contained enough that pasting it into a file and exporting it is faster than hunting for someone else's draft. Here is the most direct approach. Generate an n-by-n matrix with random values:
n = 5; A = randn(n); The semicolon suppresses the output to the command window. Without it, MATLAB prints every element, which fills your screen fast when n gets past about twenty. Solving a linear system with that matrix is one line:
x = A \ b; This is the backslash operator, and it is the single most important thing in the language. It automatically selects between LU decomposition, QR factorization, Cholesky, and a few other algorithms depending on the properties of A. You do not need to know which one it picked. You just need to make sure your matrix actually has the properties the selected algorithm expects. For eigenvalues and singular values, the commands are different and easy to mix up. eig(A) returns eigenvalues. svd(A) returns singular values. People routinely paste eig into a problem that requires SVD and then wonder why their condition number analysis is wrong. The two are related but they answer different questions. Eigenvalues tell you about the scaling along invariant directions for a square matrix. Singular values tell you about the scaling in every direction, including for rectangular matrices, and they are always non-negative.
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I spent three days debugging a simulation last year because I was using inv(A)*b instead of A\b on a condition-10^12 matrix. The two operations produced results that differed by roughly 1e-8 in relative norm, which looked fine until I fed the solution into a second stage of the pipeline and the error compounded. The fix was just swapping the operator. MATLAB's backslash estimates the condition number internally and falls back to a more stable algorithm when the matrix is ill-conditioned. inv() does not do that. It inverts the matrix directly and hands you garbage if the condition number is bad. If you need the actual inverse for some reason, like computing a fundamental matrix in control theory or generating a covariance update, use inv(A). But compute it once and reuse it. Do not call inv(A)*b repeatedly inside a loop. The overhead is negligible for a single call, and the accuracy loss is what kills you. Symbolic matrices are another trap. sym('A', [n n]) creates a named symbolic matrix, but almost nobody needs that. If you are working with symbolic expressions, define the entries individually or use sym(zeros(n)) and fill it. The named-matrix form gives you a placeholder with unknown properties, and symbolic solvers make assumptions that are not always valid. I ran into this when I tried to compute a symbolic inverse of a 3x3 parameterized matrix and got a result that looked correct but had an uncancelled common factor in the denominator. Simplifying the expression before inverting would have caught it in seconds.
Memory usage grows with n squared, which sounds obvious but people forget when n crosses a few thousand. A double-precision 5000-by-5000 matrix is about 200 MB. That is fine. A 50000-by-50000 matrix is 20 GB. Your machine will swap, and the job will take hours. Use sparse storage when your matrix is mostly zeros. A = sprandn(n, n, 0.01) creates a sparse matrix with about one percent density. Operations on sparse matrices are not automatically faster, but they do not blow up your memory the way full matrices do. One more thing about output formatting. format long shows fifteen digits. format short e shows scientific notation with four digits. The default is short fixed-point, which rounds aggressively and makes it look like your matrix has exact zeros where there are really tiny numbers. If you are debugging a matrix that should be sparse but appears dense, switch to format long first. It saved me from rewriting a preprocessing function that was filtering values below 1e-15 when the actual threshold should have been 1e-12. The code snippets above are plain enough to paste into any MATLAB script. Save the file, run it, and print the command window to PDF if you need a record. There is no magic in the syntax, and nobody is hiding a special square-matrix mode you are missing. The difficulty is in choosing the right solver for the properties of your matrix and not fighting the language by forcing operations that have better alternatives.
