Two and Four Dimensional Numbers
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To do this problem you need to know how to add and multiply two by two matrices. This video shows you an example. It might be helpful to know a little bit about Complex Numbers. You might like to read the NRICH article What are Complex Numbers? and explore Complex Numbers with the NRICH feature Adventures with Complex Numbers. |
- Let $C^*$ be the set of $2 \times2$ matrices of the form $$ \left( \begin{array}{cc} x& -y\\ y& x\end{array} \right) $$ where $x$ and $y$ are real numbers.
- Add and multiply the matrices $$\pmatrix {x & -y \cr y & x} {\rm \ and }\ \pmatrix {u & -v \cr v & u}.$$
- Consider also the subset $R^*$ for which $y=0$. Investigate addition and multiplication of matrices from $R^*$ and compare to the arithmetic of real numbers.
- Compare the arithmetic of $C^*$ with that of complex numbers.
- The matrix $\pmatrix {-1 & 0 \cr 0 & -1}$ is equivalent to the real number -1. Find a matrix equivalent to $i = \sqrt {-1}$.
This is asking you to find a matrix ${\bf M}$ which gives ${\bf M}^2=\pmatrix {-1 & 0 \cr 0 & -1}$. Use the general form of matrices in $C^*$ given at the start of this question to help you find ${\bf M}$.
- Complex numbers are two-dimensional numbers $x + iy$ where $x$ and $y$ are real numbers and $i = \sqrt {-1}$. Quaternions are four-dimensional numbers of the form $a+{\bf i}x+{\bf j}y+{\bf k}z$ where $a, x, y, z$ are real numbers and ${\bf i, j, k}$ are all different square roots of $-1$.
One model for the system of quaternions is the set of linear combinations of 2 by 2 matrices: $${\bf I}a +{\bf i}x + {\bf j}y + {\bf k}z\ = \pmatrix {1 & 0\cr 0 & 1}a+ \pmatrix {i & 0\cr 0 & -i}x+\pmatrix {0 & 1\cr -1 & 0}y+ \pmatrix {0 & i\cr i & 0}z$$ where $a, x, y, z$ are real numbers.
- Work out the matrix products: ${\bf i^2, \ j^2}$ and ${\bf k^2}$ showing that these matrices give models of three different square roots of -1.
- Work out the matrix products: ${\bf i j}$, ${\bf j i}$, ${\bf j k}$, ${\bf k j}$, ${\bf k i}$ and ${\bf i k}$.
- Investigate the sequence: ${\bf i},\ $ ${\bf i j},\ $ ${\bf i j}$ ${\bf k},\ $ ${\bf i j}$ ${\bf k i},\ $ ${\bf i j}$ ${\bf k i}$ ${\bf j,\ ...}$
- Work out the matrix products: ${\bf i^2, \ j^2}$ and ${\bf k^2}$ showing that these matrices give models of three different square roots of -1.
There are more matrix problems in this feature.
NOTES AND BACKGROUND
For one dimensional numbers (or the real numbers), each multiplication apart from multiplying by $0$ can be undone by an "inverse" (so the inverse of multiplying by $2$ is multiplying by $\frac 1 2$, or dividing by $2$ as we otherwise know it!)
For two dimensional numbers (complex numbers) we can also find the inverse of a multiplication, i.e. we can divide by a complex number. This works for all complex numbers except for $0+0i$.
Complex numbers can be used to describe points in a plane, so it felt natural to use a three dimensional number to describe points in 3D space. Unfortunately there is no way to undo a three dimensional multiplication - if you think about scalar products and vector products with 3D vectors, there is no way to uniquely undo this.
On 16th October 1843 Irish mathematician William Rowan Hamilton suddenly realised that he could find a way of multiplying four dimensional numbers and he carved his formula for doing so into the wall of Broome Bridge, Dublin. The picture at the top of this page shows the plaque on the bridge celebrating this eureka moment.
Amongst other things, Quaternions are used in orbital mechanics and computer graphics and simulations, as they can be used to describe 3D rotations without suffering from "gimbal lock". Here is a video explaining gimbal lock affected Apollo 13 (the "fourth gimbal" discussed as a solution is equivalent to quaternions being used in computer animations.) Instead of an "Euler Rotation", where three numbers are used to describe the angle of rotation in three separate directions, with Quaternion rotation three numbers are used to describe a vector in the direction of the axis of rotation and the last number represents the angle turned about this axis.
This 3Blue1Brown YouTube video introduces quaternions and shows a way of visualising them (this is quite a long video!). You might also be interested in this 3Blue1Brown YouTube video which discusses how complex numbers can be used for 2D rotations and quaternions can be used for 3D rotations.
To find out more explore these Plus articles: Curious Quaternions and Ubiquitous Octonions
Do you think of complex numbers as very abstract and strange, and the idea of 4-dimensional numbers as even stranger? If you can add and multiply two by two matrices then you will see here how two by two matrices provide a model for complex numbers with a two by two matrix corresponding to i (the square root of -1).
Two by two matrices also provide a model for the 4-dimensional numbers called quaternions and for the many different square roots of -1 that occur in this number system. Read on...
Here is another excellent solution from Andrei (Bucharest, Romania).
(1) (a) First we shall add and multiply the two matrices, obtaining: $$\pmatrix {x & -y \cr y & x} + \pmatrix {u & -v \cr v & u} = \pmatrix {x+u & -y-v)\cr y+v & x+u}$$ and $$\pmatrix {x & -y \cr y & x} \pmatrix {u & -v \cr v & u} = \pmatrix {xu-yv & -xv-yu\cr xv+yu & xu-yv}.$$ [Note the similarities here to the addition and multiplication of complex numbers.]
(b) By simple calculations we observe that $\pmatrix {0 & 0 \cr 0 & 0}$ is the identity for addition and $\pmatrix {1 & 0 \cr 0 & 1}$ is the identity for multiplication. The inverses for addition and multiplication are obtained from the conditions that
i) addition of the given matrix with its inverse gives the identity for addition, so the inverse of $\pmatrix {x & -y \cr y & x}$ for addition is: $\pmatrix {-x & y \cr -y & -x}$ and
ii) multiplying the given matrix with its inverse gives the identity for multiplication; the inverse is ${1\over (x^2+y^2)}\pmatrix {x & y \cr -y & x}.$ We see that both identity matrices and both inverses are from the set C*.
(c) Here, we shall consider R* as the set of matrices of the form $\pmatrix {x & 0 \cr 0 & x}$. These matrices could be written as $x\pmatrix {1 & 0 \cr 0 & 1} = xI_2$. Evidently $\pmatrix {x & 0 \cr 0 & x} \pm \pmatrix {y & 0 \cr 0 & y} = \pmatrix {x\pm y & 0 \cr 0 & x\pm y} = (x\pm y)I_2$.
The identity for addition is $\pmatrix {0 & 0 \cr 0 & 0}$ and the inverse of $\pmatrix {x & 0 \cr 0 & x}$ is $\pmatrix {-x & 0 \cr 0 & -x}$, which are both from R*.
For multiplication $\pmatrix {x & 0 \cr 0 & x}\pmatrix {y & 0 \cr 0 & y} =x\pmatrix {1 & 0 \cr 0 & 1}\cdot y\pmatrix {1 & 0 \cr 0 & 1} = \pmatrix {xy & 0 \cr 0 & xy}$.
The multiplicative identity is $\pmatrix {1 & 0 \cr 0 & 1}$ and the multiplicative inverse of $\pmatrix {x & 0 \cr 0 & x}$ is $\pmatrix {{1\over x} & 0 \cr 0 & {1\over x}}$.
The distributive law of addition and multiplication is the same as that of real numbers: $$\pmatrix {x & 0 \cr 0 & x}\cdot \left(\pmatrix {y & 0 \cr 0 & y}+ \pmatrix {z & 0 \cr 0 & z}\right)= \pmatrix {x & 0 \cr 0 & x}\cdot \pmatrix {y & 0 \cr 0 & y} + \pmatrix {x & 0 \cr 0 & x}\cdot \pmatrix {z & 0 \cr 0 & z} = \pmatrix {x(y+z) & 0 \cr 0 & x(y+z)}.$$ This proves that the arithmetic of R* is the same as the arithmetic of real numbers.
(d) Considering 2 complex numbers, $x + iy$ and $u + iv$:
$(x + iy) + (u + iv) = (x + u) + i(y+v)$
$(x + iy) (u + iv) = (xu - yv) + i(xv + yu).$
By simple computation we can show that multiplication in the set of two by two matrices C* is commutative and that the distributive law holds in C*. This is so because the laws apply to every operation on the components. We observe that addition and multiplication in C* are the same as addition and multiplication of complex numbers.
(e) Note that $\pmatrix {0 & -1 \cr 1 & 0}^2 = \pmatrix {-1 & 0 \cr 0 & -1}$ so we see that C* contains a model for $\sqrt -1$, the complex number $i$.
The matrix $\pmatrix {x & -y \cr y & x}$ corresponds to the complex number $x + iy$. Multiplying this number by $i$, we obtain $-y + ix$, i.e. the matrix $\pmatrix {-y & -x \cr x & -y}$. We shall consider that matrix $\pmatrix {a & b \cr c & d}$ corresponds to $i$: So, $\pmatrix {x & -y \cr y & x}\pmatrix {a & b \cr c & d} =\pmatrix {-y & -x \cr x & -y}$. The only solution for this equation is $\pmatrix {0 & -1 \cr 1 & 0}$.
A different approach is the following: we associate the point $(x,y)$ in the complex plane with the complex number $(x + iy)$ and also with the matrix $\pmatrix {x & -y \cr y & x}$. The geometrical significance of the multiplication by $i$ of a complex number is the counterclockwise rotation of the point by $\pi /2$ so $(x,y) \to (-y,x)$. In matrix notation this corresponds to $$\pmatrix {0 & -1 \cr 1 & 0}\pmatrix {x & -y \cr y & x} =\pmatrix {-y & -x \cr x & -y}.$$ (2)(a)Working out the squares of the matrices $B= \pmatrix {i & 0\cr 0 & -i}$ , $C= \pmatrix {0 & 1\cr -1 & 0}$ and $D=\pmatrix {0 & i\cr i & 0}$ we get $B^2=C^2=D^2=\pmatrix{-1 & 0\cr 0 & -1}$. So all these three matrices are square roots of -1.
(b) Similarly $BC=D=-CB$, (c)$ CD=B=-DC$ (d) $DB=C=-BD$ so these matrices are models of the quaternions $i, j$ and $k$. Then $$a\pmatrix {1 & 0\cr 0 & 1}+b\pmatrix {i & 0\cr 0 & -i} +c\pmatrix {0 & 1\cr -1 & 0}+d\pmatrix {0 & i\cr i & 0}$$ provides a model for the quaternion number system. Simple calculations show that all the field axioms hold except t
(3) Now, we shall calculate $i, i j, i j k, i j k i,...$ and we shall try to find a pattern. Using the relationships already established: $ij = k$, $ijk = k^2 = -1$, $ijki = -i$, $ijkij = -ij = -k$, $ijkijk = -kj = i$.
This means that the sequence is periodic with period 6, namely $i, k, -1, -i, -k, 1, i, k ...$.
This problem starts by using matrices as a model for complex numbers and showing that the structure
$$ \left( \begin{array}{cc} 1& 0\\ 0& 1\end{array} \right)x+ \left( \begin{array}{cc} 0& -1\\ 1& 0\end{array} \right)y $$ behaves in the same way as $x+{\text i}y$.
The idea is then extended to introduce three different two by two matrices which all square to give $$ \left( \begin{array}{cc} -1& 0\\ 0& -1\end{array} \right) $$.At the end of the problem, there are some links and videos for further information about quaternions.
There are more matrix problems in this feature.