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# Matrices and Magic Squares in Matlab

This image is filled with mathematical symbolism, and if you look carefully,

you will see a matrix in the upper right corner. This matrix is known as a

magic square and was believed by many in DÃ¼rer’s time to have genuinely

magical properties. It does turn out to have some fascinating characteristics

worth exploring.

Entering Matrices

The best way for you to get started with MATLAB is to learn how to handle

matrices. Start MATLAB and follow along with each example.

You can enter matrices into MATLAB in several different ways:

• Enter an explicit list of elements.

• Load matrices from external data files.

• Generate matrices using built-in functions.

• Create matrices with your own functions and save them in files.

Start by entering DÃ¼rer’s matrix as a list of its elements. You only have to

follow a few basic conventions:

• Separate the elements of a row with blanks or commas.

• Use a semicolon, ; , to indicate the end of each row.

• Surround the entire list of elements with square brackets, [ ].

To enter DÃ¼rer’s matrix, simply type in the Command Window

A = [16 3 2 13; 5 10 11 8; 9 6 7 12; 4 15 14 1]

MATLAB displays the matrix you just entered:

A =

16 3 2 13

5 10 11 8

9 6 7 12

4 15 14 1

This matrix matches the numbers in the engraving. Once you have entered

the matrix, it is automatically remembered in the MATLAB workspace. You

can refer to it simply as A. Now that you have A in the workspace, take a look

at what makes it so interesting. Why is it magic?

sum, transpose, and diag

You are probably already aware that the special properties of a magic square

have to do with the various ways of summing its elements. If you take the

sum along any row or column, or along either of the two main diagonals,

you will always get the same number. Let us verify that using MATLAB.

The first statement to try is

sum(A)

MATLAB replies with

ans =

34 34 34 34

When you do not specify an output variable, MATLAB uses the variable ans,

short for answer, to store the results of a calculation. You have computed a

row vector containing the sums of the columns of A. Each of the columns has

the same sum, the magic sum, 34.

How about the row sums? MATLAB has a preference for working with the

columns of a matrix, so one way to get the row sums is to transpose the

matrix, compute the column sums of the transpose, and then transpose the

result. For an additional way that avoids the double transpose use the

dimension argument for the sum function.

MATLAB has two transpose operators. The apostrophe operator (e.g., A')

performs a complex conjugate transposition. It flips a matrix about its main

diagonal, and also changes the sign of the imaginary component of any

complex elements of the matrix. The dot-apostrophe operator (e.g., A.'),

transposes without affecting the sign of complex elements. For matrices

containing all real elements, the two operators return the same result.

So

A'

produces

ans =

16 5 9 4

3 10 6 15

2 11 7 14

13 8 12 1

and

sum(A')'

produces a column vector containing the row sums

ans =

34

34

34

34

The sum of the elements on the main diagonal is obtained with the sum and

the diag functions:

diag(A)

produces

ans =

16

10

7

1

and

sum(diag(A))

produces

ans =

34

The other diagonal, the so-called antidiagonal, is not so important

mathematically, so MATLAB does not have a ready-made function for it.

But a function originally intended for use in graphics, fliplr, flips a matrix

from left to right:

sum(diag(fliplr(A)))

ans =

34

You have verified that the matrix in DÃ¼rer’s engraving is indeed a magic

square and, in the process, have sampled a few MATLAB matrix operations.

The following sections continue to use this matrix to illustrate additional

MATLAB capabilities.

Subscripts

The element in row i and column j of A is denoted by A(i,j). For example,

A(4,2) is the number in the fourth row and second column. For the magic

square, A(4,2) is 15. So to compute the sum of the elements in the fourth

column of A, type

A(1,4) + A(2,4) + A(3,4) + A(4,4)

This subscript produces

ans =

34

but is not the most elegant way of summing a single column.

It is also possible to refer to the elements of a matrix with a single subscript,

A(k). A single subscript is the usual way of referencing row and column

vectors. However, it can also apply to a fully two-dimensional matrix, in

which case the array is regarded as one long column vector formed from the

columns of the original matrix. So, for the magic square, A(8) is another way

of referring to the value 15 stored in A(4,2).

If you try to use the value of an element outside of the matrix, it is an error:

t = A(4,5)

Index exceeds matrix dimensions.

Conversely, if you store a value in an element outside of the matrix, the size

increases to accommodate the newcomer:

X = A;

X(4,5) = 17

X =

16 3 2 13 0

5 10 11 8 0

9 6 7 12 0

4 15 14 1 17

The Colon Operator

The colon, :, is one of the most important MATLAB operators. It occurs in

several different forms. The expression

1:10

is a row vector containing the integers from 1 to 10:

1 2 3 4 5 6 7 8 9 10

To obtain nonunit spacing, specify an increment. For example,

100:-7:50

is

100 93 86 79 72 65 58 51

and

0:pi/4:pi

is

0 0.7854 1.5708 2.3562 3.1416

Subscript expressions involving colons refer to portions of a matrix:

A(1:k,j)

is the first k elements of the jth column of A. Thus:

sum(A(1:4,4))

computes the sum of the fourth column. However, there is a better way to

perform this computation. The colon by itself refers to all the elements in

a row or column of a matrix and the keyword end refers to the last row or

column. Thus:

sum(A(:,end))

computes the sum of the elements in the last column of A:

ans =

34

Why is the magic sum for a 4-by-4 square equal to 34? If the integers from 1

to 16 are sorted into four groups with equal sums, that sum must be

sum(1:16)/4

which, of course, is

ans =

34

The magic Function

MATLAB actually has a built-in function that creates magic squares of almost

any size. Not surprisingly, this function is named magic:

B = magic(4)

B =

16 2 3 13

5 11 10 8

9 7 6 12

4 14 15 1

This matrix is almost the same as the one in the DÃ¼rer engraving and has

all the same “magic” properties; the only difference is that the two middle

columns are exchanged.

To make this B into DÃ¼rer’s A, swap the two middle columns:

A = B(:,[1 3 2 4])

This subscript indicates that—for each of the rows of matrix B—reorder the

elements in the order 1, 3, 2, 4. It produces:

A =

16 3 2 13

5 10 11 8

9 6 7 12

4 15 14 1

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