BASICS

Alicia Y. Tsai; Authors: Laurent EI Ghaoui; Contributors: Phan Hoang, Tony Tin, Ha To Tam, Nguyen Van Kep, Le Dinh Nam, Juliette Decugis, Nguyen Quoc Cuong.; Giuseppe C. Calafiore

9 BASICS

9.1. 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.

A matrix 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 an $m$ -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 $\mathbb{R}^{1 \times n}$ .

9.2. Transpose

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

9.3. Matrices as collections of rows

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

$B = \begin{pmatrix} b_{1}^{T} \\[0.5em] b_{2}^{T} \\[0.5em] \vdots \\[0.5em] b_{m}^{T} \end{pmatrix}.$

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

Example 1: Consider the $3 \times 2$ matrix

$A = \begin{pmatrix} 3 & 4.5 \\ 2 & 1.2 \\-0.1 & 8.2 \end{pmatrix}.$

The matrix can be interpreted as the collection of two column vectors: $A=(a_{1}, a_{2})$ , where $a_{j}$ ‘s contain the columns of $A$ :

$a_{1}=\begin{pmatrix} 3\\ 2\\-0.1\end{pmatrix},\quad \quad a_{2}=\begin{pmatrix} 4.5\\ 1.2\\8.2\end{pmatrix}.$

Geometrically, $A$ represents $2$ points in a $3$ -dimensional space.

Alternatively, we can interpret $A$ as a collection of 3-row vectors in $\mathbb{R}^2$ .

$A = \begin{pmatrix} b_{1}^{T}\\[0.5em]b_{2}^{T}\\[0.5em]b_{3}^{T} \end{pmatrix}.$

where $b_{i}$ ‘s, $i=1, 2, 3$ contain the rows of $A$ :

$b_{1}=\begin{pmatrix} 3\\ 4.5\end{pmatrix},\quad \quad b_{2}=\begin{pmatrix} 2\\ 1.2\end{pmatrix},\quad \quad b_{3}=\begin{pmatrix} -0.1\\ 8.2\end{pmatrix}.$

Geometrically, $A$ represents 3 points in a 2-dimensional space.

See also:

9.4. 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.

License

Icon for the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License

9.1. Matrices as collections of column vectors

9.2. Transpose

9.3. Matrices as collections of rows

9.4. Sparse Matrices

License

Share This Book