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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

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.

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Nicholas M. Short, Sr. email: [email protected]
Dr. Jon W. Robinson ([email protected])

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