程序代写代做代考 Section 8.7

Section 8.7
Week 8
Ali Mousavidehshikh
Department of Mathematics University of Toronto
Ali Mousavidehshikh
Week 8

Outline
1 Section 8.7
Section 8.7
Week 8
Ali Mousavidehshikh

Section 8.7
Let i2 = −1. A complex number is a number of the form z = a + ib.
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Ali Mousavidehshikh

Section 8.7
Let i2 = −1. A complex number is a number of the form z = a + ib.
The conjugate of a complex number is z = a − bi.
Week 8
Ali Mousavidehshikh

Section 8.7
Let i2 = −1. A complex number is a number of the form z = a + ib.
The conjugate of a complex number is z = a − bi. We denote the set of all complex numbers by C.
Week 8
Ali Mousavidehshikh

Section 8.7
Let i2 = −1. A complex number is a number of the form z = a + ib.
The conjugate of a complex number is z = a − bi.
We denote the set of all complex numbers by C.
All the theorems proved so far for real vector spaces remain true for complex vector spaces (this is when the scalars are complex numbers instead of real numbers).
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Section 8.7
Standard inner product in Cn: Let
z = (z1,z2,…,zn),w = (w1,w2,…,wn), where zi,wi ∈ C. We define the standard inner product ⟨z,w⟩ = 􏰀ni=1 ziwi. Notice that if z , w ∈ Rn , then ⟨z , w ⟩ = z · w .
Let z = (3, 1 − 2i , 3 + i , 2i ) and w = (1 − i , −1, −i + 1, i ). Compute zw, z/w, ⟨z, w⟩, ⟨w, z⟩, ⟨z, z⟩, and ⟨w, w⟩.
Theorem: Let z,z1,w,w1 ∈ Cn, and λ ∈ C. Then, (1) ⟨z + z1 , w ⟩ = ⟨z , w ⟩ + ⟨z1 + w ⟩, and
⟨z , w + w1 ⟩ = ⟨z , w ⟩ + ⟨z , w1 ⟩,
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Section 8.7
Norm or length of z is defined to be ∥ z ∥= 􏰟⟨z,z⟩.
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Section 8.7
Norm or length of z is defined to be ∥ z ∥= 􏰟⟨z,z⟩. Find the norm of z = (1, 1 − i , 2 + 3i , −3i ).
Week 8
Ali Mousavidehshikh

Section 8.7
Norm or length of z is defined to be ∥ z ∥= 􏰟⟨z,z⟩. Find the norm of z = (1, 1 − i , 2 + 3i , −3i ).
Theorem: For any z ∈ Cn,
(1) ∥ z ∥≥ 0, and ∥ z ∥= 0 if and only if z = 0,
Week 8
Ali Mousavidehshikh

Section 8.7
Norm or length of z is defined to be ∥ z ∥= 􏰟⟨z,z⟩. Find the norm of z = (1, 1 − i , 2 + 3i , −3i ).
Theorem: For any z ∈ Cn,
(1) ∥ z ∥≥ 0, and ∥ z ∥= 0 if and only if z = 0,
(2) ∥ λz ∥= |λ| ∥ z ∥ for all complex numbers λ (where |λ| is the norm of λ).
Avectoru∈Cn iscalledaunitvectorif∥u∥=1. Thevector
u= z isaunitvectorforanynon-zeroz∈Cn. ∥z∥
Week 8
Ali Mousavidehshikh

Section 8.7
Norm or length of z is defined to be ∥ z ∥= 􏰟⟨z,z⟩. Find the norm of z = (1, 1 − i , 2 + 3i , −3i ).
Theorem: For any z ∈ Cn,
(1) ∥ z ∥≥ 0, and ∥ z ∥= 0 if and only if z = 0,
(2) ∥ λz ∥= |λ| ∥ z ∥ for all complex numbers λ (where |λ| is the norm of λ).
Avectoru∈Cn iscalledaunitvectorif∥u∥=1. Thevector u= z isaunitvectorforanynon-zeroz∈Cn.
∥z∥
A matrix A = [aij ] is called a complex matrix if aij is a complex number for all i,j. We define its conjugate to be A = [aij ]. Notice that A + B = A + B and AB = AB .
Ali Mousavidehshikh
Week 8

Section 8.7
The transpose conjugate of A is denoted by AH, that is, AH =(A)T =AT.
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Ali Mousavidehshikh

Section 8.7
The transpose conjugate of A is denoted by AH, that is, AH =(A)T =AT.
A = A if and only if A is a real matrix.
Week 8
Ali Mousavidehshikh

Section 8.7
The transpose conjugate of A is denoted by AH, that is, AH =(A)T =AT.
A = A if and only if A is a real matrix.
Observe that AH = AT if and only if A is a real matrix.
Week 8
Ali Mousavidehshikh

Section 8.7
The transpose conjugate of A is denoted by AH, that is, AH =(A)T =AT.
A = A if and only if A is a real matrix.
Observe that AH = AT if and only if A is a real matrix.
􏰑3 2i−4+i1−i􏰒 Example: LetA= −3+i 1+i −4 2i . FindA
and AH.
Theorem: (1) (AH )H = A, (2)(A+B)H =AH +BH,
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hermitian matrix.
Section 8.7
Hermitian matrices are easy to recognize because the entries on the main diagonal must be real, and the reflection of each non-diagonal entry in the main diagonal must be the
3 −i2+i conjugate of that entry: A =  i 1 −6  is a
2−i −6 −4
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Ali Mousavidehshikh

hermitian matrix.
⟨Az,w⟩ = ⟨z,Aw⟩ for all z,w ∈ Cn.
Section 8.7
Hermitian matrices are easy to recognize because the entries on the main diagonal must be real, and the reflection of each non-diagonal entry in the main diagonal must be the
3 −i2+i conjugate of that entry: A =  i 1 −6  is a
2−i −6 −4
An n × n complex matrix A is hermitian if and only if
Week 8
Ali Mousavidehshikh

Section 8.7
Hermitian matrices are easy to recognize because the entries on the main diagonal must be real, and the reflection of each non-diagonal entry in the main diagonal must be the
3 −i2+i conjugate of that entry: A =  i 1 −6  is a
2−i −6 −4
An n × n complex matrix A is hermitian if and only if
hermitian matrix.
⟨Az,w⟩ = ⟨z,Aw⟩ for all z,w ∈ Cn.
If A is an n × n complex matrix, then λ is an eigenvalue of A if Ax = λx holds for some non-zero column x ∈ C. In this case x is called an eigenvector corresponding to λ. The characteristic polynomial cA(x) is defined by
cA(x) = det(xI − A). This polynomial has complex coefficients. All the proofs from the real case carry over.
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Ali Mousavidehshikh

Section 8.7
Theorem: Let A denote a hermitian matrix. (1) The eigenvalues of A are real,
Week 8
Ali Mousavidehshikh

Section 8.7
Theorem: Let A denote a hermitian matrix.
(1) The eigenvalues of A are real,
(2) Eigenvectors of A corresponding to distinct eigenvalues are orthogonal (in general they are linearly independent).
The definition of orthogonal sets and orthonormal sets are the same as in Rn.
Theorem: The following are equivalent for an n × n square matrix A.
(1) A is invertible and A−1 = AT .
Week 8
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Section 8.7
Schur’s Theorem: If A is any n × n complex matrix, there exists a unitary matrix U such that UHAU = T, where T is upper triangular. Moreover, the entries on the main diagonal of T are the eigenvalues of A (including multiplicities).
An n × n complex matrix A is called unitary diagonalizable if UHAU is diagonal for some unitary matrix U.
Let A be an n×n complex matrix, and let λ1,λ2,…,λn denote the eigenvalues of A, including multiplicity. Then detA = λ1λ2 ···λn and tr A = 􏰀ni=1 λi.
Schur’s Theorem tells us that every n × n complex matrix can 􏰑1 1􏰒
be unitarily triangularized. Notice that A = 0 1 cannot
be unitarily diagonalized. That is, there does not exist a unitary matrix U such that U−1AU is a diagonal matrix (to the students: try doing this).
Ali Mousavidehshikh
Week 8

Section 8.7
Spectral Theorem: If A is hermitian, then A is unitarily diagonalizable.
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Section 8.7
Spectral Theorem: If A is hermitian, then A is unitarily diagonalizable.
􏰑3 2+i􏰒
Example: LetA= 2−i 7 . NoticethatU is
hermitian. Find a unitary matrix U such that UHAU is a diagonal matrix.
Solution. The characteristic polynomial of A is
CA(x) = det(xI − A) = (x − 2)(x − 8) (both real as expected, as A is hermitian). The eigenvectors corresponding to 2 and 8
􏰑2+i􏰒 􏰑 1 􏰒
are −2 and 2 − i (orthogonal as expected),
√
U = √6 −1 2 − i . Then U is unitary (rows/columns are
respectively. Each of these eigenvectors has norm 1􏰑2+i 1􏰒
6. Letting
H 􏰑20􏰒 orthonormal) and U AU = 0 8 .
Week 8
Ali Mousavidehshikh

Section 8.7
In the real case, we know a matrix is orthogonally diagonalizable if and only if it is symmetric. The spectral theorem is the analog of half of this result (hermitian implies unitarily diagonalizable). However, unlike the real case, the
􏰑0 1􏰒 converse fails for complex matrices. The matrix −1 0 is
non-hermitian. However, it is unitarily diagonalizable. Definition: A square complex matrix N is called normal if
NNH =NHN.
Theorem: A square complex matrix N is diagonalizable if and only if N is normal.
Week 8
Ali Mousavidehshikh

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