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 coeficients
that are used to pre multiply x' to produce y, the the fourth
row. The mean vector is:
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@epix.net
Dr. Jon W. Robinson (robinson@ltpmail.gsfc.nasa.gov)