# Introductory Linear Algebra/Matrices

## Motivation

editOne important application for matrices is solving systems of linear equations. Some of the following definitions may be viewed as 'designed for solving system of linear equations'.

## Some terminologies

edit **Definition.**

(Matrix)
A matrix (plural: matrices) is a rectangular array of numbers. A horizontal unit is a *row*, and a vertical unit is a *column*.
The element in the th *row* and the th *column* is the th *entry* of the matrix.

An (read 'm by n') is a matrix with rows and columns, and is the *size* of the matrix. The rows are counted from the top, and the columns are counted from the left.
If the size of a matrix is , we simply refer to this matrix as a *number*, and no brackets are needed in this case.
The *set* of all matrices with *real entries* is denoted by .
A *capital* letter is usually used to denote a *matrix*, while *small* letters are used to denote its *entries*.
For example, denotes an matrix with entries in which
and .
(We may omit the subscript specifying the size of matrix if its size is already mentioned, or its size is not important.)

**Definition.**
(Matrix equality)
Two matrices and are *equal* if

- ,
- and
- for each pair .

We write if and are *equal*.

**Remark.**

- In other words, if two matrices have the
*same*size and*same*entries, then they are equal. - If and are
*not*equal, we write .

**Exercise.**
Consider the following three matrices and .

**Exercise.**
Let be a matrix in which each entry .
Write down in the form of an array of numbers.

(a21 a22)

(a31 a32)

In particular, if a matrix has the same number of rows and columns, then it has some nice properties. In view of the shape of such a matrix (square-like),
we define such matrices as *square matrices*.

**Definition.**
(Square matrix)
A *square matrix* is a matrix with the same number of rows and columns.

We will also introduce a term, namely *main diagonal*, which will be useful in some situations.

**Definition.**
(Main diagonal)
The *main diagonal* of an matrix (which is a *square* matrix) is
the *collection* of the th, th, , th entries.

**Example.**
The main diagonal of the matrix
is the collection of and .

**Remark.**
The matrix is the *identity matrix* (which will be defined later).

Then, we will define some types of matrices for which the definitions are related to the *main diagonal*.

**Definition.**
(Triangular matrix)
A *triangular matrix* is an *upper triangular matrix* or a *lower triangular matrix* (inclusively).

An *upper triangular matrix* is a *square* matrix whose entries *below* its *main diagonal* are all .

A *lower triangular matrix* is a *square* matrix whose entries *above* its *main diagonal* are all .

**Remark.**

- Equivalently and symbolically, a matrix is
*upper triangular*if whenever ,

and is *lower triangular* if if .

- Upper triangular and lower triangular matrices are in the form of

respectively, in which is an *arbitrary* entry (which may or may not be zero).

**Definition.**
(Diagonal matrix)
A *diagonal matrix* is a *square* matrix whose entries *not* lying on
the *main diagonal* are all .

**Remark.**

- A diagonal matrix is
*both*upper triangular and lower triangular. - A diagonal matrix has the form

in which is an *arbitrary* entry.

**Exercise.**

The last terminology we mention here is *submatrix*, which will sometimes be used.

**Definition.**
(Submatrix)
Let be a matrix. A *submatrix* of is a matrix obtained from
by *removing* some *rows* or *columns* (inclusively).

**Remark.**
By convention, every matrix is a *submatrix* of itself.

**Exercise.**

## Matrix operations

editIn this section, we will cover different matrix operations. Some operations are quite different from that in the number system, in particular, matrix multiplication.

**Definition.**
(Matrix addition and subtraction)
Let and be two matrices of the *same size*.
We define matrix *addition* and *subtraction* by

**Definition.**
(Scalar multiplication of matrix)
Let be a matrix. We define the *scalar multiplication* of the matrix by

Then, we are going to define *matrix multiplication*, which is quite different from the multiplication in the number system.

**Definition.**

(Matrix multiplication)
Let and be two matrices.
The *matrix product* of and is defined as the
matrix whose th entry is
If the number of columns of ( ) is *different* from the number of rows of ( ), then the product is *not* defined.

On the other hand, a positive *power* of a *square matrix* is defined quite similarly to that in number system.

**Definition.**
(Positive power of a square matrix)
Let be a *square* matrix. The th power of , written ,
in which is a positive number, is the *product* of *copies* of , i.e.,

**Exercise.**

Then, we will discuss matrix analogs for the numbers zero and one in the number system, namely the *zero matrix* and the *identity matrix*, which, in the number system, are analogous to the numbers and respectively.

**Definition.**
(Zero matrix)
The *zero matrix* is the matrix whose entries are all , and is denoted by or simply if there is no ambiguity.

**Remark.**
The zero matrix is analogous to the number in the number system, because:

- We have for each matrix of the same size as the zero matrix.
- We have if the products are well-defined, for each matrix .

**Definition.**
(Identity matrix)
The *identity matrix*, denoted by or simply if there is no ambiguity, is the
*diagonal matrix* whose diagonal entries are all .

**Remark.**
The identity matrix is analogous to the number in the number system, because if the products are well-defined, for each matrix .

**Example.**

- the
*zero matrix*is - the
*identity matrix*is

**Proposition.**
(Properties of matrix operations)
Let and be matrices such that the following operations are well-defined, and let be a scalar. Then, the following hold.

(i) (associativity of matrix multiplication) .

(ii) (roles of numbers zero and one)

(iii) (distributivity of matrix multiplication)

(iv) .

**Remark.**
Matrix multiplication is *not commutative* in general, i.e., the matrix product is different from the matrix product
in general.

Then, we will introduce an operation that does not exist in the number system, namely transpose.

**Definition.**
(Matrix transpose)
Let be a matrix. The *transpose* of matrix is the matrix

**Remark.**
We can see from the definition that the *transpose* of matrix is simply itself.

**Example.**

Let be the matrix . Then,

**Proposition.**
(Properties of matrix transpose)
Let and be matrices such that the following operations are well-defined.
Then, the following hold.

(i) (self-invertibility)

(ii) (linearity) for each real number and

*(iii) ('reverse multiplicativity')*

**Definition.**

(Symmetric matrix)
A matrix is a *symmetric matrix* if

**Definition.**
(Skew-symmetric matrix)
A matrix is a *skew-symmetric matrix* if

**Proposition.**
(Necessary condition for a symmetric and a skew-symmetric matrix)
A symmetric matrix or a skew-symmetric matrix *must be* a *square matrix*.

**Proof.**
It follows from observing that matrix transpose has the same size as the original matrix *if and only if* the matrix is *square matrix*,
since the number of rows and number of columns are swapped for the matrix transpose, and the size remains unchanged if and only if the number of rows equals number
of columns.

**Remark.**
This does *not* imply that every square matrix is a symmetric matrix or a skew-symmetric matrix.
The condition that a matrix is a square matrix itself is *not sufficient* (but *necessary*) for it to be symmetric or skew-symmetric.

**Example.**
is symmetric,
while is skew-symmetric,
and its transpose is . In a skew-symmetric matrix, all entries lying on the main diagonal must be 0.