is
characteristic of the multivariate discriminant analysis.
The above supermatrix contains two predictor variables and
two group coding vectors.
Other
prototypes, combining binary, continuous, and coding
variables are possible, defining data matrices for various
experimental designs and such methods of data analysis as,
e.g., analysis of covariance.
Introduction to Matrix Algebra
The summation
notation used to describe algorithms for univariate and
bivariate methods of statistical analysis has to be
exchanged for the matrix
algebra notation to describe algorithms of the
multivariate methods of data analysis. The use of matrix
algebra notation is a sine qua non within this framework, as it is
concise and avoids the plethora of subscripts, superscripts,
underbars, overbars, ellipses, carets, tildes, and other
embellishments that have to be employed when summation
notation is used to describe complex structures.
Matrices
A matrix
is a rectangular arrangement of numbers. The numbers,
contained by the matrix, are referenced by row and column
coordinates. Consider a matrix A
written in formal notation as
Subscripts of
Elements
The subscripts of the elements of the
above matrix are arranged in the row-column sequences. In
general, an element of a matrix is subscripted as
where
i is the row counter and j is the column counter.
Dimensions
The number
of rows of a matrix is typically signified by the letter n
and the number of columns by the letter k. An element of a
matrix thus can be located along the 1...i...n and the
1...j...k continua. The n and k dimensions of a matrix
determine its size.
The dimensions of the above matrix are 2 and 3.
Vectors
and
Scalars
Vectors
Standard notation for vectors is a bold
lowercase letter. The convention to denote both vectors and
matrices by an uppercase letter is specific to this book.
A vector is a special case of a matrix with either a single
row, called a row vector,
or a single column, called a column vector.
Scalars
A
matrix with a single row and a single column has only a
single element.
This element is a scalar
number. The ordinary algebra deals with scalars
and is sometimes referred to as a scalar
algebra.
Nomenclature
of Matrices
Square
Matrices
A special case of a rectangular
matrix
is a square
matrix with equal number of rows and columns.
Order
The dimension of a square matrix is
called an order. The above square matrix has its order equal
to 2.
Principal,
Supra, Infra Diagonal
Elements in a Square Matrix
Principal diagonal elements in the above square matrix is 1
and 4. The supra diagonal element is 2 and the infra
diagonal element is 3.
Symmetric
Matrices
A symmetric
matrix is a special case of square matrix with identical
supra and infra diagonal elements
Skew
Symmetric Matrices
A skew
symmetric matrix is a symmetric matrix with supra and
infra diagonal elements of equal magnitude, but opposite
sign
Diagonal
Matrices
A diagonal
matrix has all off diagonal elements equal to zero
Scalar
Matrices
A special case of a diagonal matrix is a scalar
matrix where all principal diagonal elements are equal
Identity
Matrices
A special case of a scalar matrix is an identity
matrix
with all
principal diagonal elements equal to one
and all off-diagonal elements equal to zero.
Unit
and Null Matrices
Matrices whose elements equal to one are called the unit
matrices
and the matrices filled with zeroes are
called the null
matrices
If either the row or column vectors are classified in
the above manner, a special case of a vector is
a scalar vector [2
2 2 2 2] with identical elements. Special case of a scalar
vector are a unit vector
[1 1 1 1 1] with all elements equal to one and a null
vector [0 0 0 0 0] with all elements equal to zero.
Matrix
Definitions
Transpose
A matrix definition most frequently
encountered is called the transpose
of a matrix, designated by the letter of the original matrix
with attached prime sign. The transpose of a column vector is
a row vector. The transpose of a matrix is a matrix with
interchanged rows and columns. Consider a matrix
its transpose
is obtained by writing rows of matrix A
as columns in matrix A'.
Triangulate
Another type of a matrix definition is a triangulation
of a matrix. Square symmetrical matrices, as, e.g.,
matrices of inter-correlations
are frequently
triangulated to either
supra-diagonal
or
infra-diagonal matrices
prior to publication of results of
correlational analyses to avoid repetitions of identical
values. A matrix can be also triangulated into its upper
triangular
and lower triangular
parts. A special case of triangulation
of a matrix is conversion of the negative elements of a skew
symmetric matrix
into zero elements
or a definition of a matrix
by folding it along its principal
diagonal part and subtracting its corresponding
elements (3-2=1).
Partition
Partitioning of a supermatrix into its component
submatrices can be also classified as a matrix definition.
If a matrix contains several distinct groups of elements, it
is called a supermatrix. A supermatrix contains two or more
submatrices. Consider a supermatrix
containing two
submatrices. The above
supermatrix can be partitioned into its component
submatrices
and
The partitioning of supermatrices can
be reversed and a supermatrix can be catenated
from its component submatrices. In the above example, a
supermatrix A could have been catenated from the submatrices
and
.
Augment
A matrix can be also augmented
by attaching a row or column vector to a matrix or by
attaching another matrix adjacent to the matrix row or
column margins. To augment a matrix
by a row vector [5 6] results in
augmented matrix
Decompose
The matrix structure can be decomposed
so its elements can be treated as a single variable. The
decomposition of the column structure of the matrix A
results in the row vector
The decomposition of the row structure
results in a column vector
The decomposition of both the row and
column structure of a matrix results in an unordered set of
numbers {1,2,3,4}.
Characteristics
of a Matrix
Determinant
For every square matrix, there is a
unique number called determinant,
denoted as |A|.
The determinant of a two-by two matrix
equals the product of the elements in
the principal diagonal minus the product of the off-diagonal
elements
For example,
The determinant of a matrix determines whether a matrix is invertible. We
will discuss matrix inversion in the next chapter.
Computation of determinants of square matrices larger than
two-by-two is more complicated and should be done by using
computer. An alternative notation for the determinant is
detA. For the above example we could have written that detA
= -2.
Sum
of Elements
Another unique characteristics of a matrix are sums
of its elements, either all, or sums of the elements located
in either of the parts of the matrix. The main parts of the
matrix are its rows, columns, diagonals, and the areas above
and below its principal diagonal. Another unique
characteristics of a matrix are sums of squares within these
areas.
and the column-wise sum of its elements
is a row vector
In statistical computing, these sums of
the matrix elements are often done with elements squared
before being summed. The results of these operations are
sums of squares of the matrix elements.
Matrix
Transformations
The row or column elements of a matrix
can be subjected to linear, curvilinear, logarithmic, area,
and other types of transformations.
Linear
Transformation
Consider a prototypical
data matrix
The elements of the above matrix can be
transformed into deviation scores
by subtracting the arithmetic mean of
each column of the matrix X from its constituent elements.
The matrix D can be transformed to the matrix of standard
scores Z
by dividing the elements of the matrix
D by the standard deviation of their respective columns.
Analogous operations can be also performed on matrix rows.
Normalized
Vectors
The elements of a matrix can be also normalized.
Consider a vector [.6 .8]. This vector is considered to be
normal or of a unit
length, since the sum of its squared elements equals
1.00. In formal language, a vector is normal if the scalar
product of the normalized vector with itself equals one. The
scalar products of vectors will be discussed in the chapter
to follow. To normalize a vector X, divide its each element
by a constant c
The column normalizing constants of the
matrix X equal 7.416 and the normalized matrix X, designated
as N, equals
Aside of matrix definitions and
transformations, the third broad category of matrix algebra
is that which describes operations on matrices, to be
discussed in the chapter to follow.
