mscroggs.co.uk
mscroggs.co.uk

subscribe

Blog

 2020-02-06 
This is the third post in a series of posts about matrix methods.
Yet again, we want to solve \(\mathbf{A}\mathbf{x}=\mathbf{b}\), where \(\mathbf{A}\) is a (known) matrix, \(\mathbf{b}\) is a (known) vector, and \(\mathbf{x}\) is an unknown vector.
In the previous post in this series, we used Gaussian elimination to invert a matrix. You may, however, have been taught an alternative method for calculating the inverse of a matrix. This method has four steps:
  1. Find the determinants of smaller blocks of the matrix to find the "matrix of minors".
  2. Multiply some of the entries by -1 to get the "matrix of cofactors".
  3. Transpose the matrix.
  4. Divide by the determinant of the matrix you started with.

An example

As an example, we will find the inverse of the following matrix.
$$\begin{pmatrix} 1&-2&4\\ -2&3&-2\\ -2&2&2 \end{pmatrix}.$$
The result of the four steps above is the calculation
$$\frac1{\det\begin{pmatrix} 1&-2&4\\ -2&3&-2\\ -2&2&2 \end{pmatrix} }\begin{pmatrix} \det\begin{pmatrix}3&-2\\2&2\end{pmatrix}& -\det\begin{pmatrix}-2&4\\2&2\end{pmatrix}& \det\begin{pmatrix}-2&4\\3&-2\end{pmatrix}\\ -\det\begin{pmatrix}-2&-2\\-2&2\end{pmatrix}& \det\begin{pmatrix}1&4\\-2&2\end{pmatrix}& -\det\begin{pmatrix}1&4\\-2&-2\end{pmatrix}\\ \det\begin{pmatrix}-2&3\\-2&2\end{pmatrix}& -\det\begin{pmatrix}1&-2\\-2&2\end{pmatrix}& \det\begin{pmatrix}1&-2\\-2&3\end{pmatrix} \end{pmatrix}.$$
Calculating the determinants gives $$\frac12 \begin{pmatrix} 10&12&-8\\ 8&10&-6\\ 2&2&-1 \end{pmatrix},$$ which simplifies to
$$ \begin{pmatrix} 5&6&-4\\ 4&5&-3\\ 1&1&-\tfrac12 \end{pmatrix}.$$

How many operations

This method can be used to find the inverse of a matrix of any size. Using this method on an \(n\times n\) matrix will require:
  1. Finding the determinant of \(n^2\) different \((n-1)\times(n-1)\) matrices.
  2. Multiplying \(\left\lfloor\tfrac{n}2\right\rfloor\) of these matrices by -1.
  3. Calculating the determinant of a \(n\times n\) matrix.
  4. Dividing \(n^2\) numbers by this determinant.
If \(d_n\) is the number of operations needed to find the determinant of an \(n\times n\) matrix, the total number of operations for this method is
$$n^2d_{n-1} + \left\lfloor\tfrac{n}2\right\rfloor + d_n + n^2.$$

How many operations to find a determinant

If you work through the usual method of calculating the determinant by calculating determinants of smaller blocks the combining them, you can work out that the number of operations needed to calculate a determinant in this way is \(\mathcal{O}(n!)\). For large values of \(n\), this is significantly larger than any power of \(n\).
There are other methods of calculating determinants: the fastest of these is \(\mathcal{O}(n^{2.373})\). For large \(n\), this is significantly smaller than \(\mathcal{O}(n!)\).

How many operations

Even if the quick \(\mathcal{O}(n^{2.373})\) method for calculating determinants is used, the number of operations required to invert a matrix will be of the order of
$$n^2(n-1)^{2.373} + \left\lfloor\tfrac{n}2\right\rfloor + n^{2.373} + n^2.$$
This is \(\mathcal{O}(n^{4.373})\), and so for large matrices this will be slower than Gaussian elimination, which was \(\mathcal{O}(n^3)\).
In fact, this method could only be faster than Gaussian elimination if you discovered a method of finding a determinant faster than \(\mathcal{O}(n)\). This seems highly unlikely to be possible, as an \(n\times n\) matrix has \(n^2\) entries and we should expect to operate on each of these at least once.
So, for large matrices, Gaussian elimination looks like it will always be faster, so you can safely forget this four-step method.
Previous post in series
This is the third post in a series of posts about matrix methods.
×3      ×3      ×3      ×2      ×2
(Click on one of these icons to react to this blog post)

You might also enjoy...

Comments

Comments in green were written by me. Comments in blue were not written by me.
 Add a Comment 


I will only use your email address to reply to your comment (if a reply is needed).

Allowed HTML tags: <br> <a> <small> <b> <i> <s> <sup> <sub> <u> <spoiler> <ul> <ol> <li> <logo>
To prove you are not a spam bot, please type "rebmun" backwards in the box below (case sensitive):

Archive

Show me a random blog post
 2024 

Feb 2024

Zines, pt. 2

Jan 2024

Christmas (2023) is over
 2023 
▼ show ▼
 2022 
▼ show ▼
 2021 
▼ show ▼
 2020 
▼ show ▼
 2019 
▼ show ▼
 2018 
▼ show ▼
 2017 
▼ show ▼
 2016 
▼ show ▼
 2015 
▼ show ▼
 2014 
▼ show ▼
 2013 
▼ show ▼
 2012 
▼ show ▼

Tags

tmip mathsteroids mathslogicbot golden ratio boundary element methods hats european cup craft mean mathsjam pizza cutting game show probability draughts approximation arithmetic chebyshev triangles rhombicuboctahedron live stream game of life pac-man matrix of cofactors runge's phenomenon zines manchester pi guest posts latex braiding dataset statistics advent calendar menace pi approximation day gaussian elimination sobolev spaces matrix of minors palindromes stickers youtube golden spiral gerry anderson light errors puzzles asteroids pascal's triangle noughts and crosses matrices accuracy wave scattering realhats error bars bubble bobble dragon curves ucl data visualisation christmas sorting reddit datasaurus dozen big internet math-off royal baby computational complexity games geogebra the aperiodical speed binary estimation harriss spiral preconditioning signorini conditions football coins crochet ternary radio 4 plastic ratio numbers newcastle news phd 24 hour maths martin gardner frobel inverse matrices sport bodmas go trigonometry curvature python programming machine learning finite group talking maths in public propositional calculus gather town bempp cross stitch edinburgh national lottery london underground fence posts fonts inline code christmas card convergence turtles exponential growth manchester science festival cambridge chalkdust magazine a gamut of games javascript flexagons geometry graph theory matrix multiplication fractals captain scarlet folding paper reuleaux polygons chess world cup probability oeis finite element method platonic solids interpolation databet wool pythagoras nine men's morris rugby royal institution crossnumber logs misleading statistics logo simultaneous equations squares quadrilaterals hexapawn weather station final fantasy stirling numbers london numerical analysis electromagnetic field dinosaurs map projections correlation folding tube maps data logic hannah fry dates people maths books polynomials sound weak imposition countdown tennis video games raspberry pi matt parker graphs determinants recursion standard deviation php anscombe's quartet hyperbolic surfaces

Archive

Show me a random blog post
▼ show ▼
© Matthew Scroggs 2012–2024