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,
bars, ellipses, carets, tildes, and other
embellishments that have to be employed when summation
notation is used to describe complex structures.
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 A is subscripted as aij, where
i is the row index and j is the column index.
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.
In the Matrix Module of the Visual
Statistics Studio, select ( Data Prototypes, Matrices ) and modify
the (preset) Matrix Name and Dimensions, as
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.
In the Matrix Module of the Visual
Statistics Studio, select ( Data Prototypes, Vectors ) and
double click the input box for the Marginal Referents to remove the Ent, Atr
labels. Also
modify
the (preset) Vector Name and Dimensions, as
Follow the same procedure and,
for the column vector. However, mark its corresponding box.
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. To enter a scalar number into the Visual Statistics Studio,
in the Matrix Module, select ( Data Prototypes,
Scalars), as
Nomenclature
of Matrices
Square
Matrices
A special case of a rectangular
matrix
is a square
matrix with equal number of rows and columns.
The dimension of a square matrix is
called an order. The above square matrix has its order equal
to 2.
In the Visual Statistics Studio click
on the Matrix command and select ( Data Prototypes, Matrices ).
Edit the input dialog as follows
and enter the prototype into a Matrix Cell. Click on the Red Square
command.
Symmetric
Matrices
A symmetric
matrix is a special case of square matrix with identical
supra and infra diagonal elements
In the Visual Statistics Studio,
repeat the above steps and click
on the Fold command.
Click on the Fold a Square Matrix into a Symmetric Matrix and the Red
Square commands.
Skew
Symmetric Matrices
A skew
symmetric matrix is a symmetric matrix with supra and
infra diagonal elements of equal magnitude, but opposite
sign
In the Visual Statistics Studio,
obtain a symmetric matrix as described above, and click the Fold command.
Clicking on Symmetric to Skew
Symmetric commands under the Upper and Lower labels, you can
choose the shape of the Skew Symmetric matrix.
Diagonal
Matrices
A diagonal
matrix has all off diagonal elements equal to zero
If you have in the Visual Statistics Studio the Matrix A still in a Memory Cell,
click on the Diagonalize command. Select the input matrix and click on
the Diagonal Matrix command.
Scalar
Matrices
A special case of a diagonal matrix is a scalar
matrix where all principal diagonal elements are equal
In the above dialog menu, enter 4 into the Scalar input box, and click on
the Scalar Matrix command.
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. Clicking the Identity Matrix
on the above menu, returns an Identity matrix
Unit
and Null Matrices
Matrices whose elements equal to one are called the unit
matrices.
On the Matrix Module display, click Matrices and fill out the
dialog box as
The matrices filled with zeroes are
called the null
matrices
On the Matrix Module display, click Matrices and fill out the
dialog box as
Unit
and Null Vectors
A special case of a vector is
a scalar vector [2
2 2 2 2] with identical elements.
On the Matrix Module display, click Vectors and fill out the
dialog box as
Special case of a scalar
vector are a unit vector
[1 1 1 1 1] with all elements equal to one
On the Matrix Module display, click Vectors and fill out the
dialog box as
To generate a
null
vector [0 0 0 0 0], with all elements equal to zero.
On the Matrix Module display, click Vectors and fill out the
dialog box as
