Page 96 - Matrix Analysis & Applied Linear Algebra

90 Chapter 3 Matrix Algebra

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In and , the graphs of linear functions are lines and planes through
the origin, and there seems to be a pattern forming. Although we cannot visualize
higher dimensions with our eyes, it seems reasonable to suggest that a general
linear function of the form
f(x 1 ,x 2 ,...,x n )= α 1 x 1 + α 2 x 2 + ··· + α n x n
somehow represents a “linear” or “flat” surface passing through the origin 0 =
n+1
(0, 0,..., 0) in . One of the goals of the next chapter is to learn how to
better interpret and understand this statement.
Linearity is encountered at every turn. For example, the familiar operations
of differentiation and integration may be viewed as linear functions. Since
d(f + g) df dg d(αf) df
= + and = α ,
dx dx dx dx dx
the differentiation operator D x (f)= df/dx is linear. Similarly,

(f + g)dx = fdx + gdx and αfdx = α fdx

means that the integration operator I(f)= fdx is linear.
There are several important matrix functions that are linear. For example,
the transposition function f(X m×n )= X T is linear because
T T T T T
(A + B) = A + B and (αA) = αA
(recall (3.2.1) and (3.2.2)). Another matrix function that is linear is the trace
function presented below.
Example 3.3.1
The trace of an n × n matrix A =[a ij ] is defined to be the sum of the entries
lying on the main diagonal of A. That is,
n

trace (A)= a 11 + a 22 + ··· + a nn = a ii .
i=1
Problem: Show that f(X n×n )= trace (X) is a linear function.

Solution: Let’s be efficient by showing that (3.3.3) holds. Let A =[a ij ] and
B =[b ij ], and write
n n

f(αA + B)= trace (αA + B)= [αA + B] ii = (αa ii + b ii )
i=1 i=1
n n n n

= αa ii + b ii = α a ii + b ii = α trace (A)+ trace (B)
i=1 i=1 i=1 i=1
= αf(A)+ f(B).