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.
If we call this matrix x, we can for example generate the fourth
row as a linear combination of the other rows like this: Where x' is the data matrix without row 4 and a is a vector of 3 coefficients 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 before calculating the Variance-Covariance matrix as vcv = xm*xmt The Variance-Covariance is: 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.
3
9
11
2
5
5
3
4
3
1
2
7
5
5
11
17
42
41
22
44
y = at*x'
3
9
11
2
5
5
3
4
3
1
2
7
5
5
11
2
1
3
6
3.2
6
33.2
-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
60
1
9
148
1
8.8
-19
-46.2
9
-19
44
131
148
-46.2
131
642.8
Nicholas M. Short, Sr.
email: nmshort@ptd.net
Dr. Jon W. Robinson (robinson@ltpmail.gsfc.nasa.gov)