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.