Basics

L. El Ghaoui

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 $n$ vectors $a_{1}, \dots, a_{n}$ in $\mathbb{R}^{m}$ we can define the $m \times n$ matrix $A$ with $a_{j}'$ s as columns:

$A= \begin{pmatrix} a_{1} & \dots & a_{n} \end{pmatrix} .$

Geometrically, $A$ represents $n$ points in a $m$ -dimensional space. The notation $\mathbb{R}^{m \times n}$ denotes the set of $m \times n$ matrices.

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

Transpose

The notation $A_{ij}$ denotes the element of $A$ sitting in row $i$ and column $j$ . The transpose of a matrix $A$ denoted by $A^{T}$ , is the matrix with $(i; j)$ element $A_{ji}$ , $i = 1, \dots, m$ and $j = 1, \dots, n$ .

Matrices as collections of rows

Similarly, we can describe a matrix in row-wise fashion: given $m$ vectors $b_{1}, \dots, b_{n}$ in $\mathbb{R}^{n}$ , we can define the $m \times n$ matrix $B$ with the transposed vectors $b_{i}^{T}$ as rows:

$B = \begin{pmatrix} b_{1}^{T} \\ b_{2}^{T} \\ \vdots \\ b_{m}^{T} \end{pmatrix}.$

Geometrically, $B$ represents $m$ points in a $n$ -dimensional space.

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

Examples:

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

Basics

Matrices as collections of column vectors

Transpose

Matrices as collections of rows

Sparse Matrices

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