Example 1: Norms of a matrix
Consider the matrix
A = 2 -2 1
-1 3 -1
2 -4 1
Compute
a) the 1-,
b) the - and
c) the Frobenius norm of A.

Solution:

a) The 1-norm is
||A||1 = | aij | ,   the maximum of the column sums

= max{ |2| + |-1| + |2|, |-2| + |3| + |-4|, |1| + |-1| + |2| } = max{ 5, 9, 4 } = 9

b) The -norm is
||A|| = | aij | , the maximum of the row sums
= max{ |2| + |-2| + |1|, |-1| + |3| + |-1|, |2| + |-4| + |2| } = max{ 5, 5, 8 } = 8

c) The Frobenius norm is
||A||Fr = ,
= ( |2|2 + |-2|2 + |1|2 + |-1|2 + |3|2 + |-1|2 + |2|2 + |-4|2 + |1|2 )
= (41) 6,40


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