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
Log-scaled axes
PhD thesis, chapter ∞

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 "tcesib" backwards in the box below (case sensitive):

Archive

Show me a random blog post
 2020 

Mar 2020

Log-scaled axes

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

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

Archive

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