Matrix Notation Appendix

Introduction

Here, we describe how symbols are used in these lessons represent vectors, matricies, and other matrix algebra entities.

Vectors

Bold-face, lower-case letters refer to vectors; and italic lower-case letters refer to vector elements. For example:

•  a and x refer to vectors a and x, respectively. 
• a_ij refers to the element in row i and column j of vector a.

General Matricies

Bold-face, capital letters refer to matrices, italic capital letters refer to matrix elements, and subscripts reveal matrix dimensions. For example: 

• A and X refer to matrices A and X, respectively. 
• A_ij refers to the element in row i and column j of matrix A. 
• A_ij refers to an i by j matrix A.

Spectial Matrix and Matrix Properties

Special matrices are represented by special notation. For example,

• A' refers to the transpose of matrix A.
• I refers to an identity matrix.
• I_n refers to an n x n identity matrix.
• 1 refers to the sum vector, a column vector having all of its elements equal to one.
• 1_n is a 1 x n sum vector.
• |A| refers to the determinant of matrix A.
• x refers to a matrix of deviation scores derived from the raw scores of matrix X.

Note: Like vectors, deviation score matrices are often denoted by a lower-case, boldface letter, such as x. This can cause confusion, but usually the meaning is clear from the context.

Elementary Operations

In many references, including this site, you will encounter a compact notation to describe elementary operations. That notation is shown below.

• R_i <--> R_j means to interchange rows i and j of a matrix.
• sR_i --> R_i means to multiply row i by s.
• sR_i + R_j --> R_j means to add s times row i to row j.
• C_i <--> C_j means to interchange columns i and j
• sC_i --> C_i means to multiply column i by s.
• sC_i + C_j --> C_j means to add s times column i to column j.

Echleon Matrices

• A_ref denotes a row echelon form of matrix A.
• A_rref denotes the reduced row echelon form of matrix A.