# Basic Notation

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Let us begin by introducing basic notation for matrices and vectors.

We’ll use $\RR$ to denote the set of real-numbers and $\CC$ to denote the set of complex numbers.

We write the space of all $m \times n$ real-valued matrices as $\RR^{m \times n}$. Each

With only a few exceptions, matrices are written as bold, capital letters. Matrix elements are written as sub-scripted, unbold letters. When clear from context,

instead, e.g. $A_{11}$ instead of $A_{1,1}$.

An short-hand notation for $\mA \in \RR^{m \times n}$ is

In class I’ll usually write matrices with just upper-case letters.

We write the set of length-$n$ real-valued vectors as $\RR^{n}$. Each

Vectors are denoted by lowercase, bold letters. As with matrices, elements are sub-scripted, unbold letters. Sometimes, we’ll write vector elements as

Usually, this choice is motivated by a particular application. Throughout the class, vectors are column vectors.

In class I’ll usually write vectors with just lower-case letters. I may try and follow the convention of underlining vectors. We’ll see.

# Operations

Transpose Let $\mA : m \times n$, then

Example $\mA = \sbmat{ 2 & 3 \\ 1 & 4 \\ 3 & -1 } \quad \mA^T = \sbmat{ 2 & 1 & 3 \\ 3 & 4 & -1 }$

Hermitian Let $\mA \in \CC^{m \times n}$, then

Example $\mA = \sbmat{ 2 & 3 \\ i & 4 \\ & 3 & -i } \quad \mA^* = \sbmat{ 2 & -i & 3 \\ 3 & 4 & i }$

Addition Let $\mA : m \times n$ and $\mB : m \times n$, then

Example $\mA = \sbmat{ 2 & 3 \\ 1 & 4 \\ 3 & -1 }, \mB = \sbmat{ 1 & -1 \\ 2 & 3 \\ -1 & 1 }$ $\mA + \mB = \sbmat{3 & -2 \\ 3 & 2 \\ 2 & 0 }$.

Scalar Multiplication Let $\mA : m \times n$ and $\alpha \in \RR$, then

Example $\mA = \sbmat{ 2 & 3 \\ 1 & 4 \\ 3 & -1 }, 5 \mA = \sbmat{ 10 & 15 \\ 5 & 20 \\ 15 & -5 }$

Matrix Multiplication Let $\mA : m \times n$ and $\mB : n \times k$, then

Matrix-vector Multiplication Let $\mA : m \times n$ and $\vx \in \RR^{n}$, then

This operation is just a special case of matrix multiplication that follows from treating $\vx$ and $\vc$ as $n \times 1$ and $m \times 1$ matrices, respectively.

Vector addition, Scalar vector multiplication These are just special cases of matrix addition and scalar matrix multiplication where vectors are viewed as $n \times 1$ matrices.

# Partitioning

It is often useful to represent a matrix as a collection of vectors. In this case, we write

where each $\va_j \in \RR^{m}$. This form corresponds to a partition into columns.

Alternatively, we may wish to partition a matrix into rows.

Here, each $\vr_i \in \RR^{n}$.

Using the column partitioning:

And with the row partitioning:

Another useful partitioned representation of a matrix is into blocks:

or

Here, the sizes “just have to work out” in the vernacular. Formally, all $\mA_{i,\cdot}$ must have the same number of rows and all $\mA_{\cdot,j}$ must have the same number of columns. This means the diagonal blocks are always square, but the off-diagonal blocks may not be.
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