The Matrix Reference Manual

Copyright © 1998-2022 Mike Brookes, Imperial College, London, UK
Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or any later version published by the Free Software Foundation; with no Invariant Sections, no Front-Cover Texts, and no Back-Cover Texts.  A copy of the license is included in the section entitled "GNU Free Documentation License".

To cite this manual use: Brookes, M., "The Matrix Reference Manual", [online], 2020. Please see the site accessibility statement here.


This manual contains reference information about linear algebra and the properties of real and complex matrices. The manual is divided into the following sections:

Format of Manual Entries

The general format of each entry is as follows:

  1. Definition of the term
  2. Outline of why it is important
  3. Geometric Interpretation
    The geometric interpretation of a matrix property or theorem is generally described for 2 or 3 dimensions. Words prefixed by + should be altered appropriately for other dimensions. Thus the word +area should be replaced by volume for 3-D spaces and by hyper-volume for larger spaces.
  4. List of properties and theorems:
  5. Links to related topics


The notation is based on the MATLAB software package; differences are notes below. All vectors are column vectors unless explicitly written as transposed.


Special Matrices


Several of the functions listed below have different meanings according to whether their argument is a scalar, vector or matrix. The form of the result is indicated by the function's typeface.

Other Web Sites


No originality is claimed for any of the material in this reference manual. The following books have in particular been very helpful:

I would like to thank the following people who have made suggestions or corrections to this website and apologise to anyone whose name I have omitted from this list: Gerard Baron, Mike Fairbank, Carlos Fernandes, Thomas Foregger, John Halleck, Olaf Kaehler, Ben Kennedy, James Ng, Kaare Brandt Petersen, Jacopo Piazzi, Walter Tackett, Martin Zimmermann.

This page is part of The Matrix Reference Manual. Copyright © 1998-2017 Mike Brookes, Imperial College, London, UK. See the file gfl.html for copying instructions. Please send any comments or suggestions to "mike.brookes" at "".
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