Square brackets are used to enclose elements of a matrix or vector. Use spaces or commas for horizontal concatenation, and semicolons or new lines to indicate vertical concatenation.
A = [ 1, 2, 3, 4, 5; 50 40 30 20 10
pi, sqrt(2), exp(1), (1+sqrt(5))/2, log(3) ]
A =
1.0000e+00 2.0000e+00 3.0000e+00 4.0000e+00 5.0000e+00 5.0000e+01 4.0000e+01 3.0000e+01 2.0000e+01 1.0000e+01 3.1416e+00 1.4142e+00 2.7183e+00 1.6180e+00 1.0986e+00
[m,n] = size(A)
m =
3
n =
5
A vector is considered to be a matrix with one singleton dimension.
x = [ 3; 3; 0; 1; 0 ]
x =
3 3 0 1 0
size(x)
ans =
5 1
Concatenated elements within brackets may be matrices for a block representation, as long as all the block sizes are compatible.
AA = [ A; A ]
AA =
1.0000e+00 2.0000e+00 3.0000e+00 4.0000e+00 5.0000e+00 5.0000e+01 4.0000e+01 3.0000e+01 2.0000e+01 1.0000e+01 3.1416e+00 1.4142e+00 2.7183e+00 1.6180e+00 1.0986e+00 1.0000e+00 2.0000e+00 3.0000e+00 4.0000e+00 5.0000e+00 5.0000e+01 4.0000e+01 3.0000e+01 2.0000e+01 1.0000e+01 3.1416e+00 1.4142e+00 2.7183e+00 1.6180e+00 1.0986e+00
B = [ zeros(3,2), ones(3,1) ]
B =
0 0 1 0 0 1 0 0 1
The dot-quote .' transposes a matrix. A single quote ' on its own performs the hermitian (transpose and complex conjugation). For a real matrix, the two operations are the same.
A'
ans =
1.0000e+00 5.0000e+01 3.1416e+00 2.0000e+00 4.0000e+01 1.4142e+00 3.0000e+00 3.0000e+01 2.7183e+00 4.0000e+00 2.0000e+01 1.6180e+00 5.0000e+00 1.0000e+01 1.0986e+00
x'
ans =
3 3 0 1 0
There are some convenient shorthand ways of building vectors and matrices other than entering all of their entries directly or in a loop. To get a row vector with evenly spaced entries between two endpoints, you have two options.
row = 1:4 % start:stop
row =
1 2 3 4
col = ( 0:3:12 )' % start:step:stop
col =
0 3 6 9 12
s = linspace(-1,1,5)' % start,stop,number
s =
-1.0000e+00 -5.0000e-01 0 5.0000e-01 1.0000e+00
Accessing an element is done by giving one (for a vector) or two index values in parentheses. The keyword end as an index refers to the last position in the corresponding dimension.
a = A(2,end-1)
a =
20
x(2)
ans =
3
The indices can be vectors, in which case a block of the matrix is accessed.
A(1:2,end-2:end) % first two rows, last three columns
ans =
3 4 5 30 20 10
If a dimension has only the index : (a colon), then it refers to all the entries in that dimension of the matrix.
A(:,1:2:end) % all of the odd columns
ans =
1.0000e+00 3.0000e+00 5.0000e+00 5.0000e+01 3.0000e+01 1.0000e+01 3.1416e+00 2.7183e+00 1.0986e+00
The matrix and vector senses of addition, subtraction, scalar multiplication, multiplication, and power are all handled by the usual symbols. If matrix sizes are such that the operation is not defined, an error message will result.
B = diag( [-1 0 -5] ) % create a diagonal matrix
B =
-1 0 0 0 0 0 0 0 -5
BA = B*A % matrix product
BA =
-1.0000e+00 -2.0000e+00 -3.0000e+00 -4.0000e+00 -5.0000e+00 0 0 0 0 0 -1.5708e+01 -7.0711e+00 -1.3591e+01 -8.0902e+00 -5.4931e+00
A*B causes an error. (We trap it here using a special syntax.)
try A*B, catch lasterr, end
disp('Error using *') % ignore this line
Error using *
disp(lasterr.message) % ignore this line
Inner matrix dimensions must agree.
A square matrix raised to an integer power is the same as repeated matrix multiplication.
B^3 % same as B*B*B
ans =
-1 0 0 0 0 0 0 0 -125
In many cases, one instead wants to treat a matrix or vector as a mere array and simply apply a single operation to each element of it. For multiplication, division, and power, the corresponding operators start with a dot.
C = -A;
A*C would be an error.
elementwise = A.*C
elementwise =
-1.0000e+00 -4.0000e+00 -9.0000e+00 -1.6000e+01 -2.5000e+01 -2.5000e+03 -1.6000e+03 -9.0000e+02 -4.0000e+02 -1.0000e+02 -9.8696e+00 -2.0000e+00 -7.3891e+00 -2.6180e+00 -1.2069e+00
The two operands of a dot operator have to have the same size---unless one is a scalar, in which case it is expanded or ``broadcast'' to be the same size as the other operand.
xtotwo = x.^2
xtotwo =
9 9 0 1 0
twotox = 2.^x
twotox =
8 8 1 2 1
Most of the mathematical functions, such as cos, sin, log, exp and sqrt, also operate elementwise on a matrix.
cos(pi*x')
ans =
-1 -1 1 -1 1