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 = a^t*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 coeficients

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*xm^t

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.