LECTURE 2: MATRICES
Reference: Pemberton & Rau Sections 11.2 and 11.3
A matrix is a rectangular array of numbers. A matrix with m rows and n columns is
called an mxn matrix. Matrices are usually denoted by upper case bold letters. The
numbers that form the matrix are called its entries. The entry in the ith row and jth
column of a matrix A is denoted by a ij and is referred to as the (i,j) entry of A.
Two matrices A and B are deemed to be equal (A=B) only if they are exactly the same,
with the same number of rows, the same number of columns and the same entries in the
same order.
As with vectors, the entries of a matrix can be any real numbers and, again as with
vectors, the term scalar is used to denote a member of the number system from which
entries of matrices are drawn.
The operations of addition and multiplication by a scalar are defined in a fashion
analogous to that for vectors. Addition of two matrices A and B can be performed when
and only when A and B have the same number of rows and the same number of columns,
and is performed entry by entry; multiplication by a scalar is also performed entry by
entry.
In particular, subtraction is defined by letting
-A = (-1)A, A - B = A + (-B)
The mxn matrix whose entries are all zero is called the mxn zero matrix and is denoted
by 0 mn , or simply 0 if there is no ambiguity about m and n.
Again as with vectors, the operations of addition and multiplication by scalars satisfy
laws similar to those of ordinary arithmetic. In particular we have the commutative law of
matrix addition
A+B=B+A
and the associative law of matrix addition
(A + B) + C = A + (B + C)
Matrix-vector multiplication
If w  [w1 w2 ... wn ] and x is an n-vector, then the row-column product wx is given by
wx  w1 x1  w2 x2  ...  wn xn
If A has m rows, then Ax is the m-vector whose components are the row-column products
of the rows of A with x. Thus A is a mapping from Rn to Rm.
Matrix multiplication
Suppose A is an mxn matrix and B an nxs matrix. Denote the rows of A by a1 , a2 ,..., am
and the columns of B by b1 , b 2 ,..., b s . Then AB is the mxs matrix whose (i,k) entry is the
row-column product aib k for all relevant i and k.
Matrix multiplication is composition of mappings.
Rules of matrix multiplication
The associative law of matrix multiplication states that
(AB)C = A(BC)
The rules
(A+B)C = AC+BC, A(B+C)= AB+AC
are known as distributive laws.
Also (A)B   (AB)  A(B) where  is a scalar.
Matrix multiplication is not commutative
In general,
AB  BA
This is not surprising when we recall that matrix multiplication is composition of
mappings.
Partitioned matrices
Provided all block products are defined, we can multiply two partitioned matrices block
by block using the same rule as for ordinary multiplication.
Exercises: Pemberton & Rau 11.2.1-11.2.7, 11.3.1-11.3.4