Stat Trek

Teach yourself statistics

Stat Trek

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.


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.