A Very Gentle Introduction to Matrices

The scene:

Maximus Bond meets Jill Bonza (who is related to our coordinator and security expert for this site Jim Bonza) somewhere deep inside Linear Spy headquarters.

Hello Max, what are you doing in Matrix World?

Hello Jill. Matrix World? I didn't know I was in Matrix World. This must be where they made the movie "Matrix"! Cool!

No Max. This place is all about the mathematics of matrices. Its nothing to do with the movie.

Matrices? I thought you said matrix.

We say one matrix, but the plural is matrices, Max.

Well, I like mathematics Jill, so, now that I'm here, why don't you tell me a bit about matrices?

Well Max , they are very useful in all sorts of areas, so it would be worth spending a bit of time finding out about them. Jim Bonza likes to use them for secret codes. Start here Max, here is the definition.

DEFINITION

A matrix is a rectangular array of items. In our context the items will typically be numbers.

EXAMPLES. A and B below are examples of matrices.

A =

1 4

3 0

-2 1

B =

1 9

-1 8

Most books write the array inside large brackets, but that is difficult here, so we will use a border right around instead.

TERMINOLOGY AND NOTATION

If a matrix has m rows and n columns we say it is an mxn matrix or that the SIZE of the matrix is mxn.
In the example above, A is a 3x2 matrix.

When m = n we say the matrix is SQUARE. In the example above, B is a square matrix and has size 2x2.

The numbers in the array are called ELEMENTS or ENTRIES of the matrix.

If we want to write a general matrix A, we denote the element in row i and column j as a_ij.
So a shorthand way of writing A is A = (a_ij) where i = 1, 2, ...m and j = 1, 2, ...n and by this we mean

A =

a₁₁ a₁₂ ... a_1n

a₂₁ a₂₂ ... a_2n

... ... ... ...

a_m1 a_m2 ... a_mn

For A in the example above, a₂₂ = 0, a₃₁ = -2. For B in the example above, b₁₂ = 9, b₂₂ = 8.

If a matrix has only one row it is called a ROW MATRIX or ROW VECTOR; If a matrix has only one column it is called a COLUMN MATRIX or COLUMN VECTOR.
EXAMPLES. C below is an example of a row matrix; D below is a column matrix.

C =

1 -2 3

D =

4

0

6

BASIC PROPERTIES AND OPERATIONS

EQUALITY

Two matrices are equalif and only if they have the same size and corresponding elements are equal.

EXAMPLE. When does

a b

c d

=

1 2

3 4

?

The two matrices have the same size. To be equal we also need

a = 1, b = 2, c = 3, d = 4.

ZERO MATRIX

A zero matrix is a matrix with all its entries zero. We denote the zero matrix by 0 if there is no confusion about size, otherwise we use 0_mxn.

EXAMPLES.

0_3x2 =

0 0

0 0

0 0

0_1x2 =

0 0

OPERATIONS

ADDITION (SUBTRACTION)

If A and B are the same size we can add (subtract) them and we do so by adding (subtracting) corresponding elements.

EXAMPLES.

(i)

1 2

3 4

+

2 -1

4 6

=

1 + 2 2 - 1

3 + 4 4 + 6

=

3 1

7 10

(ii)

1 2

3 4

+

2

4

is not defined because the matrices are different sizes.
PROPERTIES OF THE ADDITION OPERATION

If A, B, C are all mxn matrices (i.e. all the same size) then

A + B = B + A
A + (B + C) = (A + B) + C
A + 0_mxn = A.

SCALAR MULTIPLICATION

If A is any matrix and p is any real number, the product pA is the matrix obtained by multiplying each element of A by p.

EXAMPLE.

3

1 2

3 4

5 6

=

3 6

9 12

15 18

PROPERTIES OF THE SCALAR MULTIPLICATION OPERATION

If p and q are real numbers and A and B are matrices of the same size, then

p(A + B) = pA + pB
(p + q)A = pA + qA
pqA = p(qA)

TRANSPOSE OF A MATRIX

If A is an mxn matrix then the transpose of A, denoted A^T or A^t or sometimes A^', is the nxm matrix obtained by interchanging the rows and columns of A.

Thus the first row becomes the first column, the second row becomes the second column, etc.

EXAMPLE. Find the transpose of

1 2 3

4 5 6

The transpose is

1 4

2 5

3 6

PROPERTIES OF THE TRANSPOSE OPERATION

If A and B are matrices of the same size, then

(A^T)^T = A

(A + B)^T = A^T + B^T.

Epilogue

Max is looking very excited:

Jill, Jill, I thought you said this place had nothing to do with the movie 'Matrix", but isn't that Trinity over there?

Yes max that is Trinity. She helps us out sometimes. She is over in the identity section.

Is she getting an undercover identity Jill?

No Max. Even matrices have identities you know.

Why don't you go on to the tutOR page on Matrix Product to see what we mean by AB, i.e. two matrices multiplied together, and to learn about matrix identities?

I think I will, will you join me there, readers?