Page 395 - 35Linear Algebra

G.7 Determinants 395

we could fit these together as a (r + t) × (r + t) square block matrix

X Y
M = .
Z W
Matrix multiplication works for blocks just as for matrix entries:

2
X Y X Y X + Y Z XY + Y W
2
M = = 2 .
Z W Z W ZX + WZ ZY + W
Now lets specialize to the case where the square matrix X has an inverse.
Then we can multiply out the following triple product of a lower triangular,
a block diagonal and an upper triangular matrix:

−1
I 0 X 0 I X Y
ZX −1 I 0 W − ZX −1 Y 0 I
−1
X 0 I X Y
= −1
Z W − ZX Y 0 I

X Y
= −1 −1
ZX Y + Z W − ZX Y

X Y
= = M .
Z W
This shows that the LDU decomposition given in Section 7.7 is correct.

G.7 Determinants

Permutation Example

Lets try to get the hang of permutations. A permutation is a function which
scrambles things. Suppose we had

This looks like a function σ that has values

σ(1) = 3, σ(2) = 2, σ(3) = 4, σ(4) = 1 .

395