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Matrix Computations

David Gleich

Purdue University

Fall 2017

Course number CS-51500

Tuesday and Thursday, 10:30-11:45am

Location Forney B124


Basic Notation

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

Matrices

We'll use to denote the set of real-numbers and to denote the set of complex numbers.

We write the space of all real-valued matrices as . Each Sometimes, I'll write: instead. With only a few exceptions, matrices are written as bold, capital letters. Sometimes, we'll use a capital greek letter. Matrix elements are written as sub-scripted, unbold letters.
When clear from context, instead, e.g. instead of .

In class I'll usually write matrices with just upper-case letters. If you are unsure if something is a matrix or an element, raise your hand and ask, or quietly ask a neighbor.

Another notation for is
Sometimes this is nicer to write on the board.

Vectors

We write the set of length- real-valued vectors as . 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 and will try to follow the convention of underlining them.

Scalars

Lower-case greek letters are scalars.

Quick test

Identify the following:

Operations

Transpose Let , then

Example

Hermitian (Also called conjugate transpose) Let , then

Example

Addition Let and , then

Example .

Scalar Multiplication Let and , then

Example

Matrix Multiplication Let and , then

Matrix-vector Multiplication Let and , then This operation is just a special case of matrix multiplication that follows from treating and as and matrices, respectively.

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

Partitioning

It is often useful to represent a matrix as a collection of vectors. In this case, we write where each . This form corresponds to a partition into columns.

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

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 must have the same number of rows and all must have the same number of columns.