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
Gaussian elimination
This is the third post in a series of posts about matrix methods.

Similar posts

Gaussian elimination
Matrix multiplication
Close encounters of the second kind
Christmas (2020) is over

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>
To prove you are not a spam bot, please type "odd" in the box below (case sensitive):

Archive

Show me a random blog post
 2021 

May 2021

Close encounters of the second kind

Jan 2021

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

Tags

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

Archive

Show me a random blog post
▼ show ▼
© Matthew Scroggs 2012–2021