Computational Physics - Department of Physics

156 6 Linear Algebra

y=Ax=⇒yi=

n

∑

j= 1

ai jxj, (6.3)

matrix-matrix multiplication

A=BC=⇒ai j=

n

∑

k= 1

bikck j, (6.4)

transposition
A=BT=⇒ai j=bji,

and ifA∈Cn×n, conjugation results in

A=BT=⇒ai j=bji,

where a variablez=x−ıydenotes the complex conjugate ofz=x+ıy. In a similar way we
have the following basic vector operations, namely addition and subtraction

x=y±z=⇒xi=yi±zi,

scalar-vector multiplication
x=γy=⇒xi=γyi,

vector-vector multiplication (called Hadamard multiplication)

x=yz=⇒xi=yizi,

the inner or so-called dot product

c=yTz=⇒c=

n

∑

j= 1

yjzj, (6.5)

withca constant and the outer product, which yields a matrix,

A=yzT=⇒ai j=yizj, (6.6)

Other important operations are vector and matrix norms. A class of vector norms are the
so-calledp-norms
||x||p= (|x 1 |p+|x 2 |p+···+|xn|p)

(^1) p
,
wherep≥ 1. The most important are the 1, 2 and∞norms given by
||x|| 1 =|x 1 |+|x 2 |+···+|xn|,
||x|| 2 = (|x 1 |^2 +|x 2 |^2 +···+|xn|^2 )
(^12)
= (xTx)
(^12)
,
and
||x||∞=max|xi|,
for 1 ≤i≤n. From these definitions, one can derive several important relations, of which the
so-called Cauchy-Schwartz inequality is of great importance for many algorithms. For anyx
andybeing real-valued or complex-valued quantities, the innerproduct space satisfies
|xTy|≤||x|| 2 ||y|| 2 ,
and the equality is obeyed only ifxandyare linearly dependent. An important relation which
follows from the Cauchy-Schwartz relation is the famous triangle relation, which states that
for anyxandyin a real or complex, the inner product space satisfies