13
Jun 18

Matrix notation and summation

This is the beginning of a minicourse in matrix algebra.

Matrix notation

There are two kinds of problems when learning matrices: technical, aka boring, details, such as the matrix notation, and ideological, when you need to think as a mathematician. Neither techniques nor ideas will be repeated. When I tell you a new fact, you need to learn it there and then.

A matrix is a rectangular table of numbers. When we say that a matrix $A$ is of size $m$ by $n,$ it means that it has $m$ rows and $n$ columns. The first technical problem of working with matrices consists in learning the notation and counting dimensions. A full notation for a matrix $A$ of size $m\times n$ is

$A=\left(\begin{array}{ccccc}a_{11}&...&a_{1j}&...&a_{1n}\\...&...&...&...&...\\a_{i1}&...&a_{ij}& ...&a_{in}\\...&...&...&...&...\\a_{m1}&...&a_{mj}&...&a_{mn}\end{array}\right).$

Observe that matrices are denoted with uppercase letters and numbers with lowercase ones. An element of a matrix is provided with two subscripts: the first subscript indicates the row the element is in and the second subscript points to the column the element is in. Often instead of the above full notation a shorter one is used: $A=\left( a_{ij}\right) _{m\times n}.$

Two cases are of special interest: matrices with one column, called column vectors, and with one row, called row vectors. The notation for the two is

$A=\left(\begin{array}{c}a_{1}\\...\\a_{i}\\...\\a_{m}\end{array}\right)=\left(a_{i}\right)_{m\times 1},\ \ A=\left(\begin{array}{ccccc}a_{1}&...&a_{j}&...&a_{n}\end{array}\right)_{1\times n}=\left(a_{j}\right)_{1\times n}$

(obviously, for vectors generic elements can be provided with one subscript). For now enough of boring stuff.

Vector summation

Global idea 1. You know what is a matrix, when you know what operations can be performed with it. To introduce or understand those operations, look at properties of numbers and see if they have analogs for matrices. We start realizing this idea using vectors.

Intuition. Suppose a bank has $n$ customers, and each of them has two account types: a checking account, that pays no interest, and a deposit, an interest bearing account. All checking accounts can be collected in a vector $C=\left(\begin{array}{ccccc}c_{1}&...&c_{j}&...&c_{n}\end{array}\right)$ and all deposits can be put into another vector $D=\left(\begin{array}{ccccc}d_{1}&...&d_{j}&...& d_{n}\end{array}\right).$ If the bank wants to know the overall holdings of each customer, it can find them by adding the two vectors:

(1) $C+D=\left(\begin{array}{ccccc}c_{1}+d_{1}&...&c_{j}+d_{j}&...&c_{n}+d_{n}\end{array} \right)$

(adding a checking account of one customer to a deposit account of another customer wouldn't make sense). Elements of a vector are also called its components or coordinates, so (1) means component-wise summation. Note that we can do this only if the vectors $C,D$ have the same number of components.

Mathematical reasoning. Summation of numbers has the following properties:

A) Commutativity: $a+b=b+a$ for any two numbers $a,b.$

B) Associativity: $a+(b+c)=(a+b)+c$ for any three numbers $a,b,c.$

C) Existence of zero: there is a special number, denoted $0,$ such that $a+0=a$ for any number $a.$

D) Existence of an opposite number: for any number $a,$ there is another number, denoted $-a$, such that $a+(-a)=0.$

After this we say that two numbers are equal if their difference is zero: $a=b$ if and only if $a-b\equiv a+(-b)=0.$

Exercise 1. 1) Does vector summation have properties similar to A) and B)? 2) What vector would you call a zero vector? What property does it have? 3) For a given vector $A,$ how would you define its opposite $-A?$ What property does it have? 4) When two vectors can be considered equal?

Remark. When formulating properties similar to A)-D), be careful with existence ("there is") and universality ("for any") clauses.

Matrix summation

Exercise 2. 1) Before formulating properties of matrix sums, ask yourself when do they exist? In other words, under what condition on matrix dimensions we can add them? Then you can define a sum. 2) Does matrix summation have properties similar to A) and B)? 2) What matrix would you call a zero matrix? What property does it have? 3) For a given matrix $A,$ how would you define its opposite $-A?$ What property does it have? 4) When two matrices can be considered equal?