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.
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.
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.
Matrix subscripts denote echelon forms.
- Aref denotes a row echelon form of matrix A.
- Arref denotes the reduced row echelon form of matrix A.