mscroggs.co.uk
mscroggs.co.uk

subscribe

Blog

Inverting a matrix

 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
PhD thesis, chapter ∞
PhD thesis, chapter 5

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 "u" then "n" then "c" then "o" then "u" then "n" then "t" then "a" then "b" then "l" then "e" in the box below (case sensitive):

Archive

Show me a random blog post
 2020 

Feb 2020

PhD thesis, chapter ∞
PhD thesis, chapter 5
PhD thesis, chapter 4
PhD thesis, chapter 3
Inverting a matrix
PhD thesis, chapter 2

Jan 2020

PhD thesis, chapter 1
Gaussian elimination
Matrix multiplication
Christmas (2019) is over
 2019 
▼ show ▼
 2018 
▼ show ▼
 2017 
▼ show ▼
 2016 
▼ show ▼
 2015 
▼ show ▼
 2014 
▼ show ▼
 2013 
▼ show ▼
 2012 
▼ show ▼

Tags

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

Archive

Show me a random blog post
▼ show ▼
© Matthew Scroggs 2012–2020