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

-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.

Primary Author: Nicholas M. Short, Sr. email: nmshort@epix.net

Collaborators: Code 935
NASA GSFC, GST, USAF
Academy

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