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. Within the Matrix module, select the Matrix command under the Data Prototypes heading and generate matrix X

Click on the Transpose command

Matrix X' contains rows of matrix X as its columns.

Pre-multiplying a matrix by its transpose  X'X ( Operands [ 2] [ 1 ] )

 Click the X * Y command and the operands in the [ 2 ]  [ 1 ] order Operands indicating that matrix X (in the Matrix X Cell [ 1 ] is pre-multiplied by its transpose in the Matrix Cell [ 2 ] .

The result is

a (3,3), (Atr, Atr) matrix.

Post-multiplying a matrix by its transpose  XX' ( Operands [ 1] [ 2 ] )

This time, click the X * Y command again and notice the highlighted [ 1 ]  [ 2 ]  Operands, indicating that matrix X (in the Matrix X Cell [ 1 ] is post-multiplied by its transpose in the Matrix Cell [ 2 ] .

The result is

a (7,7), (Ent, Ent) matrix.

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. A matrix can be also triangulated into its upper triangular or lower triangular

parts.

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 the above supermatrix can be restored from its component submatrices by clicking the Join command. A matrix can be also augmented, reduced, or decomposed.

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.

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. The sum of elements in the principal diagonal of a matrix is called a trace. Thus the trace of a matrix X

equals 1 + 5 + 9, i.e., 15.

The sum of the supra-diagonal elements equals 11, the sum of the infra-diagonal elements equals 19. The sum of elements equals 45. The row-wise sum of its elements is a column vector 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 (the Standardize command)

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 above matrix can be transformed to the matrix of standard scores

 

by dividing the elements of the matrix of deviation scores by the standard deviation of their respective columns.