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Norm and inner products in C n ,and abstract inner product spaces

Math 130 Linear AlgebraD Joyce, Fall 2012

Weve seen how norms and inner products workin R n . They can also be dened for C n . Theres awrinkle in the denition of complex inner products.

The norm of a complex vector v. Well startwith the norm for C which is the one-dimensionalvector space C 1 , and extend it to higher dimen-sions.

Recall that if z = x + iy is a complex numberwith real part x and imaginary part y, the complexconjugate of z is dened as z = x iy , and theabsolute value, also called the norm, of z is denedas

|z | = x 2 + y2 = z z.Now, if v = ( v1 , v 2 , . . . , v n ) is a vector in C n

where each vi is a complex number, well dene its

norm v asv = (v1 , v 2 , . . . , v n )

= |v1 |2 + |v2 |2 + + |vn |2 = nk =1

|vk |2 .

Note that if the coordinates of v all happen to bereal numbers, then this denition agrees with thenorm for real vector spaces.

Norms on C n enjoy many of the same properties

that norms on Rn

do. For instance, the norm of any vector is nonnegative, and the only vector withnorm 0 is the 0 vector. Also, norms are multiplica-tive in the sense that

cv = |c| vwhen c is a complex number and v is a complexvector.

Furthermore, the triangle inequality for complexnorms holds

v w v + w .Well prove it later.

The inner product v |w of two complex vec-tors. We would like to have a complex innerproduct that (1) extends the real product, (2) isconnected to the complex norm by the equation

v 2 = v , v , and (3) has nice algebraic proper-ties such as bilinearity.

In order to get property (2), well have to intro-duce a wrinkle into the denition. We cannot de-

ne (v1 , . . . , v n )|(w1 , . . . , w n ) as v1 w1 + vn wn ,because then (v1 , . . . , v n )|(v1 , . . . , v n ) would equalv21 + v

2n which doesnt equal |v1 |

2 + + |vn |2 . If

we throw in a complex conjugate, however, it willwork. That explains the following denition.

Denition 1. The standard complex inner prod-uct of two vectors v and w in C n is dened by

v |w = (v1 , v2 , . . . , v n )|(w1 , w 2 , . . . , w n )= v1 w1 + v2 w2 +

vn wn =

n

k =1

vk wk

It follows that for each v Cn , our desired con-

dition (2) above, holds

v 2 = v |v .Also, condition (1) holds. If v and w happen to bea real vectors, then their complex inner product isthe same as their real inner product.

Most of the algebraic properties of complex inner

products are the same as those of real inner prod-uct. For instance, inner products distribute overaddition,

u |v + w = u |v + u |w ,and over subtraction,

u |v w = u |v u |w ,1

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and the inner product of any vector and the 0 vec-tor is 0

v |0 = 0 .However, complex inner products are not com-

mutative. Instead they have the property

u |v = v |u .Complex inner products are linear in their rst

argument. If c is a complex scalar, then

cu |v = c u |vIn for the second argument, we have instead

u

|cv

=c u

|v .

The complex conjugate of c comes from our deni-tion where we use the complex conjugates of coor-dinates of the second vector.

In summary, complex inner products are not bi-linear, but they are linear in the rst argument andconjugate linear in the second argument.

Abstract linear spaces. So far, weve looked atthe standard real inner product on R n and the stan-

dard complex inner product on Cn

. Although wereprimarily concerned with standard inner products,there are other inner products, and we should con-sider the generalization of these standard innerproducts. Well call a vector space equipped withan inner product an inner product space.

We can make the denitions for abstract innerproduct spaces for both the real case and the com-plex case at the same time. In the denition, welltake the scalar eld F to be either R or C .

Denition 2. An inner product space over F is avector space V over F equipped with a functionV V F that assigns to vectors v and w in V ascalar denoted v |w , called the inner product of vand w , which satises the following four conditions:

(a). u + v |w = u |w + u |w ,(b). cv |w = c v |w ,(c). v |w = w |v , and

(d). v |v > 0 if v = 0 .For an inner product space, the norm of a vector vis dened as v = v |v .

Note that when F = R , condition (c) simply saysthat the inner product is commutative.

Properties (a) and (b) state that the inner prod-uct is linear in the rst argument. Using those and(c), you can show that the inner product is conju-gate linear in the second argument.

Condition (d) says that the norm v of a vectoris always positive except in the one case the thatv = 0 . From condition (b) you can infer 0 = 0.Another property of norms is that cv = |c| v .Finally, two more properties of inner products arethe Cauchy-Schwarz inequality | v |w | v w ,and the triangle inequality v + w v + w .Well prove them later.Examples. The standard inner products on R n

and C n are, of course, the primary examples of in-ner product spaces.

Our text describes some other inner productspaces besides the standard ones R n and C n . Oneis a real inner product on the vector space of con-tinuous real-valued functions on [0 , 1]. Another isan inner product on m n matrices over either Ror C . Well discuss those briey in class. Theresanother example of the vector space of complex-valued functions on the unit circle we wont havetime for.

Math 130 Home Page athttp://math.clarku.edu/~djoyce/ma130/

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