Inverting Matrices Determinants and Matrix Multiplication Determinants Square

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 # 1 2 3 4 5 X 12 18 32 44 49 Y 1 3 2 4 5 from which the following XY variance-covariance matrix is obtained: X Y X 256 21. 5 Y 21. 5 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).

Consider This Data Matrix Subject # 1 X 10 Y 1 2 20 2 3 30 3 4 40 4 5 50 5 Variance-Covariance Matrix X Y X 250 25 Y 25 2. 5 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 2 x 2 Matrix • For our original variance/covariance matrix:

Multiplying a Scalar by a Matrix • Simply multiply each matrix element by the scalar (1/177. 75 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 256 21. 5, comma at end of row 21. 5 2. 5}; matrix within { } determinant = det(XY); find determinant inverse = inv(XY); find inverse identity = XY*inverse; multiply by inverse quit;

XY 2 rows 2 cols 256 21. 5 2. 5 DETERMINANT 1 row 1 col 177. 75 INVERSE 2 rows 2 cols 0. 0140647 -0. 120956 -0. 1209560 1. 440225 IDENTITY 2 rows 2 cols 1 -2. 22 E-16 -2. 08 E-17 1