Special Matrices

L. El Ghaoui

Special Matrices

Square Matrices
- Identity and diagonal matrices
- Triangular matrices
- Symmetric matrices
- Orthogonal Matrices
Dyads

Some special square matrices

Square matrices are matrices that have the same number of rows as columns. The following are important instances of square matrices.

Identity matrix

The $n \times n$ identity matrix (often denoted $I_{n}$ , or simply $I$ , if the context allows), has ones on its diagonal and zeros elsewhere. It is square, diagonal, and symmetric. This matrix satisfies $A \dots I_{n}=A$ for every matrix $A$ with $n$ columns, and $I_{n}\dots B = B$ for every matrix $B$ with $n$ rows.

Matlab syntax

>> I3 = eye(3); % the 3x3 identity matrix
>> A = eye(3,4); % a 3x4 matrix having the 3x3 identity in its first 3 columns

Diagonal matrices

Diagonal matrices are square matrices $A$ with $A_{ij}=0$ when $i \neg j$ . A diagonal $n \times n$ matrix $A$ can be denoted as $A = \diag(a)$ , with $a \in \mathbb{R}^{n}$ the vector containing the elements on the diagonal. We can also write

$A = \begin{pmatrix} a_{1} & & \\ & \ddots & \\ & & a_{r} \end{pmatrix},$

where by convention the zeros outside the diagonal are not written.

Matlab syntax

>> A = diag([1 2 3]); % a diagonal matrix with 1,2,3 on the diagonal
>> A = spdiags([1 2 3]',0,3,3); % the same matrix declared as a sparse matrix

Symmetric matrices

Symmetric matrices are square matrices that satisfy $A_{ij}=A_{ji}$ for every pair $(i, j)$ . An entire section is devoted to symmetric matrices.

Triangular matrices

A square matrix $A \in \mathbb{R}^{m \times n}$ is upper triangular if $A_{ij}=0$ when $i < j$ . Here are a few examples:

$A_{1} = \begin{pmatrix} 1 & -1 \\ 0 & 2 \\ 0 & 0 \end{pmatrix}, A_{2} = \begin{pmatrix} 3 & 8 & 3 \\ 0 & 6 & -1 \end{pmatrix}, A_{3} = \begin{pmatrix} 0 & 8 & 3 \\ 0 & 0 & -1 \end{pmatrix}.$

A matrix is lower triangular if its transpose is upper triangular. For example:

$A = \begin{pmatrix} 1 & 0 & 0 \\ 8 & -9 & 0 \\ 1 & -2 & 3 \end{pmatrix}.$

Orthogonal matrices

Orthogonal (or, unitary) matrices are square matrices, such that the columns form an orthonormal basis. If $U = [u_{1},\dots,u_{n}]$ is an orthogonal matrix, then

$u_{i}^{T}u_{j} = \begin{cases} 1 \text{ if $i=j$} \\ 0 \text{otherwise.}\end{cases}.$

Thus, $U^{T}U = I_{n}.$ Similarly, $UU^{T} = I_{n}.$

Orthogonal matrices correspond to rotations or reflections across a direction: they preserve length and angles. Indeed, for every vector $x$ ,

$||Ux||_{2}^{2} = (Ux)^{T}(Ux) = x^{T}U^{T}Ux=x^{T}x=||x||_{2}^{2}.$

Thus, the underlying linear map $x \rightarrow Ux$ preserves the length (measured in Euclidean norm). This is sometimes referred to as the rotational invariance of the Euclidean norm.

In addition, angles are preserved: if $x, y$ are two vectors with unit norm, then the angle $\theta$ between them satisfies $\cos \theta = x^{T}y$ , while the angle $\theta'$ between the rotated vectors $x' = Ux, y' = Uy$ satisfies $\cos \theta'=(x')^{T}y'$ . Since

$(Ux)^{T}(Uy)=x^{T}U^{T}Uy=x^{T}y,$

we obtain that the angles are the same. (The converse is true: any square matrix that preserves lengths and angles is orthogonal.)

Geometrically, orthogonal matrices correspond to rotations (around a point) or reflections (around a line passing through the origin).

Examples:

Dyads

Dyads are a special class of matrices, also called rank-one matrices, for reasons seen later.

Definition

A matrix $A \in \mathbb{R}^{m \times n}$ is a dyad if it is of the form $A = uv^{T}$ for some vectors $u \in \mathbb{R}^{m}, v \in \mathbb{R}^{n}$ . The dyad acts on an input vector $x \in \mathbb{R}^{n}$ as follows:

$Ax=(uv^{T})x=(v^{T}x)u.$

In terms of the associated linear map, for a dyad, the output always points in the same direction $u$ in output space ( $\mathbb{R}^{m}$ ), no matter what the input $x$ is. The output is thus always a simple scaled version of $u$ . The amount of scaling depends on the vector $v$ , via the linear function $x \rightarrow v^{T}x.$ .

Example: Single-factor models of financial data.

Normalized dyads

We can always normalize the dyad, by assuming that both $u, v$ are of unit (Eculidean) norm, and using a factor to capture their scale. That is, any dyad can be written in normalized form:

$A = uv^{T} = (||u||_{2}\cdot||v||_{2})\cdot \left(\dfrac{u}{||u||_{2}}\right)\left(\dfrac{v}{||v||_{2}}\right)^{T} = \sigma\tilde{u}\tilde{v}^{T},$

where $\sigma>0$ , and $||\tilde{u}||_{2}=||\tilde{v}||_{2}=1$ .

Special Matrices

Some special square matrices

Identity matrix

Diagonal matrices

Symmetric matrices

Triangular matrices

Orthogonal matrices

Dyads

Definition

Normalized dyads

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