Stat Trek

Teach yourself statistics

Teach yourself statistics

# Matrix Notation

Here, we describe how symbols are used on the Stat Trek web site to represent vectors, matrices, 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.
• ai refers to the ith element in vector a.

## General Matrices

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

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

## Special Matrices 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.
• In 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.
• 1n 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.

• Ri <--> Rj means to interchange rows i and j of a matrix.
• sRi --> Ri means to multiply row i by s.
• sRi + Rj --> Rj means to add s times row i to row j.
• Ci <--> Cj means to interchange columns i and j
• sCi --> Ci means to multiply column i by s.
• sCi + Cj --> Cj means to add s times column i to column j.

## Echelon Matrices

Matrix subscripts denote echelon forms.