Page 30 - Vector Analysis

26 CHAPTER 1. Linear Algebra

1.6.1 Matrix norms

Each m ˆ n matrix A P M(m, n; F) induces a linear map L : Fn Ñ Fm in a natural way: let

A = [aij]mˆn be a m ˆ n matrix, B = tejujn=1 and Br = trekukm=1 be the standard basis of Fn
and Fm, respectively. We define the linear map L : Fn Ñ Fm by

mn n

Lx = ÿ ÿ aijxjrei P Fm, where x = ÿ xjej P Fn ,

i=1 j=1 j=1

or equivalently, [Lx]Br = A[x]B. The linear map L is called the linear map induced by
the matrix A.

By matrix norms it means the operator norm of the induced linear map. However,

as introduced in Section 1.6, the operator norm of a linear map depends on the norms

equipped on the vector spaces. In particular, we have introduced p-norm on Fn, and we

have the following

Definition 1.88. Let A P M(m, n; F) with induced linear map L : Fn Ñ Fm. The p-norm
of A, denoted by }A}p, is the operator norm of L : (Fn, } ¨ }p) Ñ (Fm, } ¨ }p) given by

}A}p = sup }Lx}p = sup }Lx}p .
}x}p
}x}p=1 x‰0

Remark 1.89. We can also choose different p in the domain and the co-domain. In other

words, the (p, q)-norm of A P M(m, n, F) is the operator norm of the induced linear map
L : (Fn, } ¨ }p) Ñ (Fm, } ¨ }q) given by

}A}(p,q) = sup }Lx}q = sup }Lx}q .
}x}p
}x}p=1 x‰0

From now on, for notational simplicity we use Ax to denote [Lx]Br if Br is the
standard basis of the co-domain.

Example 1.90. Consider the case p = 1 and p = 8, respectively.

! mm m )
|anj| .
1. p = 8: }A}8 = sup max ÿ ÿ ÿ

}Ax}8 = |a1j |, |a2j |, ¨ ¨ ¨

}x}8=1 j=1 j=1 j=1
[]
Reason: Let x = (x1, x2, ¨ ¨ ¨ , xn)T and A = aij nˆm. Then


a11x1 + ¨ ¨ ¨ + a1mxm
 a21 x1 a2mxm 
Ax = + ¨ ¨¨ +
...

an1x1 + ¨ ¨ ¨ + anmxm