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Inner Products in
Complex Vector
Spaces

9
...

Our first thought would be to determine if we can extend the dot product to e11•
Does this define an inner product on e11? Let z x iJ, then we have

+

=

Z Z = Z1 + +
= (� +
...
- y�) +2i(X1Y1 +
...
0
does not even make sense
...
fining
an inner product in en
...
We recall that if z E e, then zZ lzl2 ?:'
...
Hence, it
makes sense to choose
...
/(z,

=

w>= z· w

as this gives us

(z,Z'J
Definition

(

In en the standard inner product , ) is defined by

Standard Inner Product on C"

(z,

EXAMPLE 1

= z· t= Z1Z1 +· · · +ZnZn = [zi[2 +· · · +[ Zn [2 ?:'
...
+ZnWn,

[ �]
...


Solution:

= v
...
(2 - i)il = r-� : � ] (2 [;: �]
= r-�:�Jc2+i)[���]
= (2+i)(3+
= +23i
i

2i)(2

i)(l - i)
4

-1

·

-1


...


- i)

Observe that this does not satisfy the properties of the real inner product
...


EXERCISE 1

Let it=

[i � ]
2i

and v=

[i � ��l

Determine (it, v), (2iil, v), and (it, 2iv)
...


Definition

Let V be a vector space over C
...


Note that property (1) allows us to define the (standard) length by llz ll = (z, z)112,
as desired
...
Notice that if all the
vectors are real, the Hermitian property simplifies to symmetry
...
However,
this property reduces to bilinearity when the scalars are all real
...
How­
ever, new proofs are required
...
Then, for all w,

z EV,
(4) l(z, w)I ::
...
:; llzll + llwll

Proof:

We prove (4) and leave the proof of (5) as Problem Dl
...

Let C(m, n) denote the complex vector space of m x n matrices with complex en­
tries
...
11111• Of course, we want to define

the standard inner product on C(m, n) in a similar way
...
Thus, we define the inner product (,)on C(m, n) by (A, B)= tr(i/ A)
...
Then

Since we want the trace of this, we just consider the diagonal entries in the product
and find that
m

tr(BT A) =

m

m

I a;1b;1 + I a;2bi2 + ··· + I a;nbi11
i=l

i=l

i=l

which corresponds to the standard inner product of the corresponding vectors under
the obvious isomorphism with C1111•

EXAMPLE2

Let

i
2
A =
[ ; 1 � i]

[ _;i

and B =

Find (A,B) and show that this corre-

+

sponds to the standard inner product of

Solution:

�: ��l
2 i
1
a=
i
1 -i

We have

�

and b=

3
2 - 3i
-2i
...


matrix with complex entries
...


Theorem 2

Let A and B be complex matrices and let

(1) (Az, w) =(z,A*w)
(2) A** =A
(3) (A B)*= A* B*

for all Z, w

+

E

+

(4) (aA)*= aA*
(5) (AB)* =B* A*

The proof is left as Problem

D2
...
Then

conjugate transpose A•

Orthogonality in c_n and Unitary Matrices
With a complex inner product defined, we can proceed and introduce orthogonality,
projections, and distance, as we did in the real case
...
However, we must be very careful that we
have the vectors in the correct order when calculating an inner product since the com­
plex inner product is not symmetric
...
, vk} is an orthonormal

...

Notice that since (z, w) may be complex, there is no obvious way to define the

angle between vectors
...
, and v3

=

-1

1 and consider the subspace S

=

Span{v1, v2, v3}

0

ofe4
...


Solution: Let w1

=

0
Then
0
0

...

(b) Let z

:
...


=

-1

EXAMPLE4

Solution: Using the orthogonal basis we found in (a), we get

(continued)

...
w1 >

i

0
-i
0
0

...


w2

2

0

+

...


llw3ll2
1
0
0
-1

-

6

w3

2

W hen working with real inner products, we saw that orthogonal matrices are very
important
...


Definition

Unitary Matrix

An n x n matrix with complex entries is said to be unitary if its columns form an
n
orthonormal basis for e
...


p T We get the associated result for unitary matrices
...


(1)

The columns of U form an orthonormal basis for IC"

(2) The rows of U form an orthonormal basis for IC"
(3) u-1
u·
=

Proof: Let U
z1
if and only if (z;, Z;)
=

Z,]
...
j

-T
( U• U) ij - Z;
_, j
=

, Zn} is orthonormal

1 and
0

we get that U* U

• • •

I if and only if {z1,

equivalent to (3) is similar
...
The proof that (2

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Title: Inner products in complex vector spaces
Description: Linear algebra course

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