An orthogonal matrix
If A is an n×nsquare matrix satisfying ATA=AA T =I, then Ais called an
orthogonal matrix , and A−^1 =AT.An example of an orthogonal matrix is given by
A=(
cos θ sin θ
−sin θ cos θ)
, A T=(
cos θ −sin θ
sin θ cos θ)
, AA T=IExample 10-9. Some examples of special matrices are:
A=
a 11 0 0 0
a 21 a 22 0 0
a 31 a 32 a 33 0
a 41 a 42 a 43 a 44 is lower triangular
B=
b 11 b 12 b 13 b 14
0 b 22 b 23 b 24
0 0 b 33 b 34
0 0 0 b 44 is upper triangular
I=
1 0 0
0 1 0
0 0 1 is an identity matrix which is also diagonal
T=
β γ 0 0 0
α β γ 0 0
0 α β γ 0
0 0 α β γ
0 0 0 α β
is a tridiagonal matrix
A=
1 0 0 0
0 0 1 0
0 1 0 0
0 0 0 1 is an orthogonal matrix satisfying AA T=I
If f=f( ̄x) = f(x 1 , x 2 ,... , x n)is a function of n-variables, then the Hessian matrix
associated with f is
H=
∂^2 f
∂x 12∂^2 f
∂x 1 ∂x 2 ···∂^2 f
∂x 1 ∂xn
∂^2 f
∂x 2 ∂x 1∂^2 f
∂x 22 ···∂^2 f
∂x 2 ∂xn..
.
..
. ...
..
.
∂^2 f
∂xn∂x 1∂^2 f
∂xn∂x 2 ···∂^2 f
∂xn^2