### Singular Matrices

A singular matrix is one in which one or more of the rows or columns can be calculated as a linear combination of the other rows or columns. If one calculates the Variance-Covariance matrix of a singular data matrix, the determinant of that Variance-Covariance matrix will be 0.

For example consider the "data" matrix below with 4 variables and 5 observations.

3 | 9 | 11 | 2 | 5
5 | 3 | 4 | 3 | 1
2 | 7 | 5 | 5 | 11
17 | 42 | 41 | 22 | 44

If we call this matrix x, we can for example generate the fourth row as a linear combination of the other rows like this:

y = at*x'

Where x' is the data matrix without row 4

3 | 9 | 11 | 2 | 5
5 | 3 | 4 | 3 | 1
2 | 7 | 5 | 5 | 11

and a is a vector of 3 coefficients

2
1
3

that are used to pre multiply x' to produce y, the the fourth row. The mean vector is:

6 | 3.2 | 6 | 33.2

We then subtract the mean vector from each "observation" to shift the mean to zero

Matrix with mean vector shifted to Zero
-3 | 3 | 5 | -4 | -1
1.8 | -0.2 | 0.8 | -0.2 | -2.2
-4 | 1 | -1 | -1 | 5
-16.2 | 8.8 | 7.8 | -11.2 | 10.8

before calculating the Variance-Covariance matrix as vcv = xm*xmt

The Variance-Covariance is:

60 | 1 | 9 | 148
1 | 8.8 | -19 | -46.2
9 | -19 | 44 | 131
148 | -46.2 | 131 | 642.8

and the determinant is: 3.699*10-11 which is within rounding error of 0

If we delete the 4 th variable and recalculate the determinat for the 3 variable data set, we get: 473.2 clearly much larger than 0! As an exercise, you can try calculating this value by hand, or with a matrix algebra package. Mathcad 5 plus was used to calculate this example.

Nicholas M. Short, Sr. email: nmshort@ptd.net
Dr. Jon W. Robinson (robinson@ltpmail.gsfc.nasa.gov)