| Basic definitions
and operations of matrix algebra - needed for multivariate analysis |
Elementary Matrix Algebra
Vectors and matrices are arrays or tables of numbers. The algebra for
symbolic operations on them is different from the algebra for operations
on scalars, or single numbers. For example there is no division in matrix
algebra, although there is an operation called "multiplying by an inverse".
It is possible to express the exact equivalent of matrix algebra equations
in terms of scalar algebra expressions, but the results look rather messy.
It can be said that the matrix algebra notation is shorthand for the corresponding
scalar longhand.
A vector is a column of numbers
The scalars ai are the elements of vector a.
The transpose of a, denoted by a' is the row arrangement
of the elements of a.
The sum of two vectors is the vector of sums of corresponding elements.
The difference of two vectors is the vector of differences of corresponding
elements.
The product a'b is a scalar formed by
which may be written in shortcut notation as
where ai and bi are the ith elements of vector a
and b respectively
The product ab' is a square matrix
The product of a scalar k, times a vector a, is k times
each element of a
A matrix is a rectangular table of numbers, with p rows and
n columns. It is also referred to as an array of n column
vectors of length p. Thus
is a p by n matrix. The typical element of A is aij,
denoting the element of row i and column j.
Matrices are added and subtracted on an element by element bases.
Thus
Matrix multiplication involves the computation of the sum of the
products of elements from a row of the first matrix (the premultiplier
on the left) and a column of the second matrix (the postmultiplier on the
right). This sum of products is computed for every combination of rows
and columns For example, if A is a 2 x 3 matrix and B
is a 3 x 2 matrix, the product AB is
Thus the product is a 2 x 2 matrix. This came about as follows: The
number of columns of A must be equal to the number of
rows of B, In this case this is 3. If they are not equal multiplication
is impossible. If they are equal then the number of rows of the product
AB is equal to the number of rows of A and the number of
columns is equal to the number of columns of B.
It follows that the result of the product BA is a 3 x 3 matrix
In general, if A is a k x p and B is a p
x n matrix, the product AB is a k x
n matrix. If k = n then the product BA can also
be formed. We say that matrices conform for the operations of addition,
subtraction or multiplication when their respective orders (numbers of
row and columns) are such as to permit the operations. matrices that do
not conform for addition or subtraction cannot be added or subtracted.
matrices that do not conform for multiplication cannot be multiplied. |
