mscroggs.co.uk
mscroggs.co.uk

subscribe

Blog

Matrix multiplication

 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
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 "z" then "e" then "r" then "o" 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

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

Archive

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