Singular Matrices¶
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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
-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 |
Table: Matrix with mean vector shifted to Zero
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.