Matrix Algebra

More Special Matrices
Permutations and Transpositions
Permutation and Transposition Matrices
Elementary Row Operations
Determinants
Determinants of Order 2
Determinants of Order 3
Characterizing the Determinant Function
Rules for Determinants
Expansion by Alien Cofactors and the Adjugate Matrix
Minor Determinants
The Inverse Matrix
Denition and Existence
Orthogonal Matrices
Denition
Given Nn = f1; : : : ; ng where n  2,
a permutation of Nn is a bijective mapping  : Nn ! Nn.
The family n of all permutations of Nn includes:
I the identity mapping  dened by (h) = h for all h 2 Nn;
I for each  2 n, a unique inverse ??1 2 n
for which ??1   =   ??1 = 
Denition
1. Given any permutation  2 n, an inversion of 
is a pair (i ; j) 2 Nn such that i > j and (i) < (j ).
2. A permutation  : Nn ! Nn is either even or odd
according as it has an even or odd number of inversions.
3. The sign or signature of a permutation , denoted by s
Lemma
For all  2 n one has sgn() =
Q
i>j s((i) ?? (j)) =
N
S().
Proof.
Let p := #f(i ; j) 2 Nn  Nn j i > j & (i) < (j)g
denote the number of inversions of .
By denition, sgn() = (??1)p = 1 according as p is even or odd.
But the denitions on the previous slide imply that
p = #f(i ; j) 2 Nn  Nn j i > j & s((i) ?? (j)) = ??1g
= #f(i ; j) 2 Nn  Nn j sij () = ??1g
Therefore
sgn() = (??1)p =
Q
i>j s((i ) ?? (j))
=
Qn
i=1
Qn
j=1 sij () =
N
S()