mscroggs.co.uk
mscroggs.co.uk

subscribe

Blog

 2020-01-23 
This is the first post in a series of posts about matrix methods.
When you first learn about matrices, you learn that in order to multiply two matrices, you use this strange-looking method involving the rows of the left matrix and the columns of this right.
It doesn't immediately seem clear why this should be the way to multiply matrices. In this blog post, we look at why this is the definition of matrix multiplication.

Simultaneous equations

Matrices can be thought of as representing a system of simultaneous equations. For example, solving the matrix problem
$$ \begin{bmatrix}2&5&2\\1&0&-2\\3&1&1\end{bmatrix} \begin{pmatrix}x\\y\\z\end{pmatrix} = \begin{pmatrix}14\\-16\\-4\end{pmatrix} $$
is equivalent to solving the following simultaneous equations.
\begin{align*} 2x+5y+2z&=14\\ 1x+0y-2z&=-16\\ 3x+1y+1z&=-4 \end{align*}

Two matrices

Now, let \(\mathbf{A}\) and \(\mathbf{C}\) be two 3×3 matrices, let \(\mathbf{b}\) by a vector with three elements, and let \(\mathbf{x}=(x,y,z)\). We consider the equation
$$\mathbf{A}\mathbf{C}\mathbf{x}=\mathbf{b}.$$
In order to understand what this equation means, we let \(\mathbf{y}=\mathbf{C}\mathbf{x}\) and think about solving the two simuntaneous matrix equations,
\begin{align*} \mathbf{A}\mathbf{y}&=\mathbf{b}\\ \mathbf{C}\mathbf{x}&=\mathbf{y}. \end{align*}
We can write the entries of \(\mathbf{A}\), \(\mathbf{C}\), \(\mathbf{x}\), \(\mathbf{y}\) and \(\mathbf{b}\) as
\begin{align*} \mathbf{A}&=\begin{bmatrix} a_{11}&a_{12}&a_{13}\\ a_{21}&a_{22}&a_{23}\\ a_{31}&a_{32}&a_{23} \end{bmatrix} & \mathbf{C}&=\begin{bmatrix} c_{11}&c_{12}&c_{13}\\ c_{21}&c_{22}&c_{23}\\ c_{31}&c_{32}&c_{23} \end{bmatrix} \end{align*} \begin{align*} \mathbf{x}&=\begin{pmatrix}x_1\\x_2\\x_3\end{pmatrix} & \mathbf{y}&=\begin{pmatrix}y_1\\y_2\\y_3\end{pmatrix} & \mathbf{b}&=\begin{pmatrix}b_1\\b_2\\b_3\end{pmatrix} \end{align*}
We can then write out the simultaneous equations that \(\mathbf{A}\mathbf{y}=\mathbf{b}\) and \(\mathbf{C}\mathbf{x}=\mathbf{y}\) represent:
\begin{align} a_{11}y_1+a_{12}y_2+a_{13}y_3&=b_1& c_{11}x_1+c_{12}x_2+c_{13}x_3&=y_1\\ a_{21}y_1+a_{22}y_2+a_{23}y_3&=b_2& c_{21}x_1+c_{22}x_2+c_{23}x_3&=y_2\\ a_{31}y_1+a_{32}y_2+a_{33}y_3&=b_3& c_{31}x_1+c_{32}x_2+c_{33}x_3&=y_3\\ \end{align}
Substituting the equations on the right into those on the left gives:
\begin{align} a_{11}(c_{11}x_1+c_{12}x_2+c_{13}x_3)+a_{12}(c_{21}x_1+c_{22}x_2+c_{23}x_3)+a_{13}(c_{31}x_1+c_{32}x_2+c_{33}x_3)&=b_1\\ a_{21}(c_{11}x_1+c_{12}x_2+c_{13}x_3)+a_{22}(c_{21}x_1+c_{22}x_2+c_{23}x_3)+a_{23}(c_{31}x_1+c_{32}x_2+c_{33}x_3)&=b_2\\ a_{31}(c_{11}x_1+c_{12}x_2+c_{13}x_3)+a_{32}(c_{21}x_1+c_{22}x_2+c_{23}x_3)+a_{33}(c_{31}x_1+c_{32}x_2+c_{33}x_3)&=b_3\\ \end{align}
Gathering the terms containing \(x_1\), \(x_2\) and \(x_3\) leads to:
\begin{align} (a_{11}c_{11}+a_{12}c_{21}+a_{13}c_{31})x_1 +(a_{11}c_{12}+a_{12}c_{22}+a_{13}c_{32})x_2 +(a_{11}c_{13}+a_{12}c_{23}+a_{13}c_{33})x_3&=b_1\\ (a_{21}c_{11}+a_{22}c_{21}+a_{23}c_{31})x_1 +(a_{21}c_{12}+a_{22}c_{22}+a_{23}c_{32})x_2 +(a_{21}c_{13}+a_{22}c_{23}+a_{23}c_{33})x_3&=b_2\\ (a_{31}c_{11}+a_{32}c_{21}+a_{33}c_{31})x_1 +(a_{31}c_{12}+a_{32}c_{22}+a_{33}c_{32})x_2 +(a_{31}c_{13}+a_{32}c_{23}+a_{33}c_{33})x_3&=b_3 \end{align}
We can write this as a matrix:
$$ \begin{bmatrix} a_{11}c_{11}+a_{12}c_{21}+a_{13}c_{31}& a_{11}c_{12}+a_{12}c_{22}+a_{13}c_{32}& a_{11}c_{13}+a_{12}c_{23}+a_{13}c_{33}\\ a_{21}c_{11}+a_{22}c_{21}+a_{23}c_{31}& a_{21}c_{12}+a_{22}c_{22}+a_{23}c_{32}& a_{21}c_{13}+a_{22}c_{23}+a_{23}c_{33}\\ a_{31}c_{11}+a_{32}c_{21}+a_{33}c_{31}& a_{31}c_{12}+a_{32}c_{22}+a_{33}c_{32}& a_{31}c_{13}+a_{32}c_{23}+a_{33}c_{33} \end{bmatrix} \mathbf{x}=\mathbf{b} $$
This equation is equivalent to \(\mathbf{A}\mathbf{C}\mathbf{x}=\mathbf{b}\), so the matrix above is equal to \(\mathbf{A}\mathbf{C}\). But this matrix is what you get if follow the row-and-column matrix multiplication method, and so we can see why this definition makes sense.
This is the first post in a series of posts about matrix methods.
Next post in series
Gaussian elimination

Similar posts

Inverting a matrix
Gaussian elimination
Happy √3-ϕ+3 Approximation Day!
A surprising fact about quadrilaterals

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 "decagon" in the box below (case sensitive):

Archive

Show me a random blog post
 2020 

Jul 2020

Happy √3-ϕ+3 Approximation Day!

May 2020

A surprising fact about quadrilaterals
Interesting tautologies

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

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

Archive

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