Inverting Matrices Determinants and Matrix Multiplication

Determinants Square matrices have determinants, which are useful in other matrix operations, especially inversion. For a second-order square matrix, A, the determinant is

Consider the following bivariate raw data matrix Subject #12345 X Y13245 from which the following XY variance-covariance matrix is obtained: XY X Y 2.5

Think of the variance-covariance matrix as containing information about the two variables – the more variable X and Y are, the more information you have. Any redundancy between X and Y reduces the total amount of information you have -- to the extent that you have covariance between X and Y, you have less total information.

Generalized Variance The determinant tells you how much information the matrix has about the variance in the variables – the generalized variance, after removing redundancy among variables. We took the product of the variances and then subtracted the product of the covariances (redundancy).

Imagine a Rectangle Its width represents information on X Its height represents information on Y X is perpendicular to Y (orthogonal), thus r XY = 0. The area of the rectangle represents the total information on X and Y. With covariance = 0, the determinant = the product of the two variances minus 0.

Imagine a Parallelogram Allowing X and Y to be correlated with one another moves the angle between height and width away from 90 degrees. As the angle moves further and further away from 90 degrees, the area of the parallelogram is also reduced. Eventually to zero (when X and Y are perfectly correlated). See the Generalized Variance video clip in BlackBoard.

Consider This Data Matrix Subject #12345 X Y12345 XY X25025 Y 2.5 Variance-Covariance Matrix Since X and Y are perfectly correlated, the generalized variance is nil.

Identity Matrix An identity matrix has 1’s on its main diagonal, 0’s elsewhere.

Inversion The inverted matrix is that which when multiplied by A yields the identity matrix. That is, AA  1 = A  1 A = I. With scalars, multiplication by the inverse yields the scalar identity. Multiplication by an inverse is like division with scalars.

Inverting a 2x2 Matrix For our original variance/covariance matrix:

Multiplying a Scalar by a Matrix Simply multiply each matrix element by the scalar (1/ in this case). The resulting inverse matrix is:

A  A  1 = A  1  A = I

The Determinant of a Third-Order Square Matrix

Matrix Multiplication for a 3 x 3

SAS Will Do It For You Proc IML; reset print; display each matrix when created XY ={ enter the matrix XY , comma at end of row }; matrix within { } determinant = det(XY); find determinant inverse = inv(XY); find inverse identity = XY*inverse; multiply by inverse quit;

XY 2 rows 2 cols DETERMINANT 1 row 1 col INVERSE 2 rows 2 cols IDENTITY 2 rows 2 cols E E-17 1