Basics

Matrices as collections of column vectors
Transpose
Matrices as collections of row vectors
Sparse matrices

Matrices as collections of column vectors

Matrices can be viewed simply as a collection of (column) vectors of same size, that is, as a collection of points in a multi-dimensional space.

Matrices can be described as follows: given

vectors

in $mathbf{R}^m$ , we can define the m times n

matrix

with

's as columns:

$A = left(begin{array}{ccc} a_1 & ldots & a_n end{array}right).$

Geometrically,

represents

points in a

-dimensional space. The notation $mathbf{R}^{m times n}$ denotes the set of

matrices.

With our convention, a column vector in $mathbf{R}^{n}$ is thus a matrix in $mathbf{R}^{n times 1}$ , while a row vector in $mathbf{R}^{n}$ is a matrix in $mathbf{R}^{1 times n}$ .

Transpose

The notation $A_{ij}$ denotes the element of

sitting in row

and column

. The transpose of a matrix

. denoted by A^T

, is the matrix with (i,j)

element $A_{ji}$ , i=1,ldots,m

Matrices as collections of rows

Similarly, we can describe a matrix in row-wise fashion: given

vectors

in $mathbf{R}^n$ , we can define the m times n

matrix

with the transposed vectors b_i^T

as rows:

$B = left( begin{array}{c} b_1^T vdots b_m^T end{array}right) .$

Matlab syntax

>> A = [1 2 3; 4 5 6]; % a 2x3 matrix
>> B = A'; % this is the transpose of matrix A
>> B = [1 4; 2 5; 3 6]; % B can also be declared that way
>> C = rand(4,5); % a random 4x5 matrix

Sparse Matrices

In many applications, one has to deal with very large matrices that are sparse, that is, they have many zeros. It often makes sense to use a sparse storage convention to represent the matrix.

One of the most common formats involves only listing the non-zero elements, and their associated locations (i,j)

in the matrix. In Matlab, the function sparse allows to create a sparse matrix based on that information. The user needs also to specify the size of the matrix (it cannot infer it from just the above information).

Matlab syntax

>> S = sparse([3 2 3 4 1],[1 2 2 3 4],[1 2 3 4 5],4,4) % creates a 4x4 matrix
>> S = [0 0 0 5; 0 2 0 0; 1 3 0 0; 0 0 4 0]; % the same matrix with ordinary convention