by Ebrahim Patel & Andrew Irving (on Tuesday 14th October 2014)

Suppose we take *x* to represent the distance I must walk from my house to work. Then 2*x* represents the total distance I would walk in one day’s journey to work.

Now, what if I also walk from work to a nearby place for lunch? Let us represent this distance by *y* so that 2*y* gives the distance form work to lunch and back.

We could describe all of my journeys in this way, e.g. walking from home to the park on a Saturday, walking from home into town to do some shopping on a Sunday.

If we list all of my possible journeys, labelling their distances as *x(1), x(2), …, x(n)* respectively (where *n* is the total number of different journeys undertaken), then these can be represented by a size *n* vector

**x** = *(x(1), x(2), …, x(n))*.

Suppose that I also want to record the total distance I walk on each day of the 7-day week. Then this can be represented by a size 7 vector

**d*** = (d(1), d(2), …, d(7))*

where *d(1)* represents the total distance walked on Monday, *d(2)* for Tuesday, and so on.

These vectors, **x** and **d**, are clearly linked. And the **link** **between them** can be described by a **matrix** *M*.

The first row of *M *will represent Monday: the number of times I take the first journey will be entered into position *M(11)*, the number of times I take the second journey is *M(12)*, and so on.

So each row of *M* represents a day of the week (giving it 7 rows) while each column represents one of my possible journeys (giving *it* *n* columns). *M* is therefore a 7 × *n* matrix.

When multiplied by the vector **x**, we get the vector **d**, i.e. *M.***x** = **d**, or

As you may know, a vector is just a list of numbers. A matrix is then just a **list of vectors**!

In fact, matrix multiplication is a lot like vector multiplication, essentially using the same mechanism – the *dot product* (or *scalar product*). Here, row *i* of **d** is equal to the dot product of **x** and row *i* of *M*.

Now that we have the basics, let us show how all this can be so useful. A new colleague wants to know the closest place for lunch – they’ve come to the right person!

Because, as I like to know if I do enough exercise, I wear a pedometer. This records my daily walking (i.e. my pedometer tells me all numerical values of **d**) whilst I record all values of *M *myself.

For my colleague, I need to find which one of my various lunch terms (in vector **x**) is the smallest – I need **x**. Well, I know that *M*.**x**= **d **but how do** **I rearrange for **x**?

If this were a scalar equation (say, *m.x = d* where *m*,* x*, and *d* are all scalars), then we would simply divide by *m* on both sides to give *x = d/m*. Put another way, we would multiply each side by the inverse of *m*.

And indeed that is the approach we take with our matrix-vector equation. Thus, we find **x** by multiplying both sides of *M*.**x**= **d **by the inverse of *M* (denoted inv(*M*)) to give,

**x** = inv(*M*).**d**

noting that the right hand side gives a vector of known numerical values, as required.* A complicated problem is thereby reduced to a simple one using matrices.

So, matrices are an extension to scalar arithmetic – they allow us to multiply, add, and solve equations for many different scenarios in one fell swoop – a short-cut for large scale arithmetic.

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* Although some matrices do not, the matrix *M *is assumed to have an inverse here.

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