inverting matrices determinants and matrix multiplication
TRANSCRIPT
Determinants
• Square matrices have determinants, which are useful in other matrix operations, especially inversion.
• For a second-order square matrix, A,the determinant is
21122211 aaaa A
2221
1211
aa
aa
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 YX 256 21.5Y 21.5 2.5
9.05.2256
5.21
YX
XY
SS
COVr
75.177)5.21(5.21)5.2(256 AThink 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 rXY = 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 MatrixSubject # 1 2 3 4 5
X 10 20 30 40 50
Y 1 2 3 4 5
X Y
X 250 25Y 25 2.5
Variance-Covariance Matrix
15.2250
25
YX
XY
SS
COVr 0)25(25)5.2(250 A
Since X and Y are perfectly correlated, the generalized variance is nil.
Inversion
• The inverted matrix is that which when multiplied by A yields the identity matrix. That is, AA1 = A1A = I.
• With scalars, multiplication by the inverse yields the scalar identity.
• Multiplication by an inverseis like division with scalars.
.11
a
a
.1
b
a
ba
Inverting a 2x2 Matrix
• For our original variance/covariance matrix:
256 21.5-
21.5- 2.5
75.177
1
-
- *
1
1121
1222
aa
aa
2
12 A
A
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:
51.44022503 .120956399-
.120956399- .014064698 1A
AA1 = A1A = I
dzcxdycw
bzaxbyaw
zy
xw
dc
ba
colrow colrow
colrow colrow
2212
2111
1 0
0 1
51.44022503 .120956399-
.120956399- .014064698
2.5 21.5
21.5 256
The Determinant of a Third-Order Square Matrix
3A
333231
232221
131211
aaa
aaa
aaa
33211223321113223121
3213312312332211
aaaaaaaaaa
aaaaaaaa
Matrix Multiplication for a 3 x 3
izhwgtiyhvgsixhugr
fzewdtfyevdsfxeudr
czbwatcybvascxbuar
zyx
wvu
tsr
ihg
fed
cba
332313
322212
312111
colrow colrow colrow
colrow colrow colrow
colrow colrow colrow
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;