Matrix Review

Matrix Operations

In the following, we will consider n x m matrices of the form:

A vector is an n x 1 matrix, for instance:

Matrix Transpose:

Example:

Symmetric Matrices

Note that a matrix A is symmetric if A' = A; that is, if aij = aij. Important examples of symmetric matrices in multivariate statistics include the variance-covariance matrix and the correlation matrix. These shall be defined when we consider descriptive statistics.

Examples:

This matrix below

is symmetric.

This matrix below

is not symmetric.

Addition

The sum of two matrices:

Here the notation "n×m" means that each of the matrices A, B, and C has n rows and m columns. Two matrices may be added if and only if they have identical numbers of rows and they have identical numbers of columns. Matrices are added by summing the corresponding columns of each matrix. Thus the ijth column of C is obtained by summing the ijth elements of A and B.

Example:

Inpsect matrix addition

Multiplication

The product of two matrices:

Here the number of columns in A must equal the number of rows in B. Note: In general, AB ≠ BA.

Example:

Inpsect matrix multiplication

The Identity Matrix

The identity matrix has ones in the diagonal and zeros in the off-diagonal elements:

It is called the identity matrix since multiplication of any matrix A by the identity matrix yields the original matrix A:

AI = IA = A

Matrix Inverse

Square matrices only: A-1 is the inverse of A if

AA-1 = I

For 2 x 2 Matrices, we have the formula:

Example:

Always check your work!

General n x n Matrices:

To obtain an algorithm for inverting general n x n matrices, we must review three elementary row operations:

  1. Exchange two rows.
  2. Multiply the elements of a row by a constant.
  3. Add a multiple of another row to the given row.

Obtaining the Inverse of Matrix A

To obtain the inverse of a n x n matrix A :

Step 1: Create the partitioned matrix ( A I ) , where I is the identity matrix.

Step 2: Perform elementary row operations on the partitioned matrix with the objective of converting the first part of the matrix to the identity matrix.

Step 3: Then the resulting partitioned matrix will take the form ( I A-1 )

Step 4: Check your work by demonstrating that AA-1 = I.

Below is a demonstration of this process:


Click on the "Next" above, to continue this lesson.

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