MATB24H3 Lecture Notes - Orthogonal Matrix, Orthogonal Complement, Hermitian Matrix

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Published on 21 Apr 2013
School
UTSC
Department
Mathematics
Course
MATB24H3
University of Toronto at Scarborough
Department of Computer & Mathematical Sciences
MAT B24S Fall 2011
MAT B24
Lecture 19-20
Matrices with Complex Numbers
In this lecture we will work with matrices that have entries from C.
Since we know that Cis a field we know that every nonzero element in C
has an inverse. And from the previous lecture we know that for an nonzero
zC, z1=z
|z|2. We use that to reduce matrices with complex numbers
as entries.
EXAMPLE: Find the inverse of the matrix A=
i1 3i
1 + i1 2i
1 0 1 + i
solution.
A=
i1 3i
1 + i1 2i
1 0 1 + i
100
010
001
iR1
1i
2R2
1R3
1i3
11i
21 + i
1 0 1i
i0 0
01i
20
0 0 1
R2R1
R3R1
1i3
01+i
22 + i
0i4i
i0 0
i1i
20
i01
(1i)R2
iR3
1i3
0 1 1 + 3i
0 1 1 + 4i
i0 0
1 + ii0
1 0 i
R3R2
1i3
0 1 1 + 3i
0 0 i
i0 0
1 + ii0
i i i
R1+3iR3
R2+(3i)R3
iR3
1i0
0 1 0
0 0 1
i+ 3 33
4i14i13i
1 1 1
R1+iR2
100
010
001
i1 1 + i i
4i14i13i
1 1 1
1
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Thus: A1=
i1 1 + i i
4i14i13i
1 1 1
Similarly,:
(i) the process of finding if a set of vectors in Cnis linearly independent
the same as the process in Rn.
(ii) given a vector vCn, the process of finding a coordinate vector vB
relative to a given ordered basis Bof Cnis also the same as the process
in Rn.
However, you must always use the addition and multiplication as defined
in the field C.
Euclidean Inner Product in Cn
We know that Cnis a vector space, but is it an inner product space? The
dot product makes Rnan inner product space. Does it make Cnalso?
If we try it we see that there are some problems.
For example, in C2if v= [a, ai] for a6= 0 gives
kvk2= [a, ai]·[a, ai] = a2a2= 0.
Also if b= [a, bi] with a < b then:
kbk2= [a, bi]·[a, bi] = a2b2<0
So we would get the magnitude of a nonzero vector to be 0 and the magnitude
of a vector to be negative. That is definitely contrary to the definition of an
inner product. So the dot product is not an inner product in Cn.
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Regardless, Cnhas an inner product defined on it:
DEFINITION: If u= [u1, u2, ..., un] and v= [v1, v2, ..., vn] are in Cnthen
the Euclidean inner product of u and v is:
<u,v>=u1v1+u2v2+... +unvn
This inner product has the following properties:
THEOREM: Let u,v,wCnand let zCbe a scalar.Then:
1. <u,u>0, and <u,u>= 0 if and only if u=0.
2. <u,v>=<v,u>,
3. <(u+v),w>=<u,w>+<v,w>,
4. <w,(u+v)>=<w,u>+<w,v>,
5. < zu,v>=z < u,v>, and <u, zv>=z < u,v>.
proof.
1. <u,u>=u1u1+u2u2+... +unun=|u1|2+|u2|2+... +|un|20
and it is equal to zero if and only if every |ui|2= 0, which is true if and
only if each ui= 0
2. <u,v>=u1v1+u2v2+... +unvn=u1v1+u2v2+... +unvn=
u1v1+u2v2+... +unvn=v1u1+v2u2+... +vnun=<v,u>
Verify the rest as an exercise.
This inner product is not commutative as the inner product is in Rnand
also, the result of an inner product in Cnis in C.
It seems like this inner product does not satisfy the inner product of
spaces over the reals. So it is puzzling why it is called an inner product. The
reason is that the definition given here of an inner product is satisfied by the
definition of an inner product over the reals.
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Document Summary

In this lecture we will work with matrices that have entries from c. Since we know that c is a eld we know that every nonzero element in c has an inverse. And from the previous lecture we know that for an nonzero z c, as entries. We use that to reduce matrices with complex numbers z 1 = z. Example: find the inverse of the matrix a = i. 2 2 + i i 4 i. However, you must always use the addition and multiplication as de ned in the eld c. The dot product makes rn an inner product space. If we try it we see that there are some problems. For example, in c2 if v = [a, ai] for a 6= 0 gives kvk2 = [a, ai] [a, ai] = a2 a2 = 0. Also if b = [a, bi] with a < b then: kbk2 = [a, bi] [a, bi] = a2 b2 < 0.

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