Transcript ENGG2013 Lecture 20 - Chinese University of Hong Kong

ENGG2013 Unit 20
Extensions to Complex numbers
Mar, 2011.
Definition: Norm of a vector
• By Pythagoras theorem, the length of a vector
with two components [a b] is
• The length of a vector with three components
[a b c] is
• The length of a vector with n components,
[a1 a2 … an], is defined as
,
which is also called the norm of [a1 a2 … an].
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Examples
• We usually denote the norm of a vector v by
|| v ||.
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Norm squared
• The square of the norm, or square of the
length, of a column vector v can be
conveniently written as the dot product
• Example
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REVIEW OF COMPLEX NUMBERS
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Quadratic equation
• When the discriminant of a quadratic
equation is negative, there is no real solution.
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• The complex roots
are
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y
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0
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x
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Complex eigenvalues
• There are some matrices whose eigenvalues
are complex numbers.
• The characteristic polynomial of this matrix is
The eigenvalues are
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Complex numbers
• Let i be the square root of –1.
• A complex number is written in the form
a+bi
where a and b are real numbers.
“a” is called the “real part” and “b” is called the
“imaginary part” of a+bi.
• Addition: (1+2i) + (2 – i) = 3+i.
• Subtraction: (1+2i) – (2 – i) = –1 + 3i.
• Multiplication: (1+2i)(2 – i) = 2+4i–i–2i2=4+3i.
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Complex numbers
• The conjugate of a+bi is defined as a – bi.
• The absolute value of a+bi is defined as
(a+bi)(a – bi) = (a2+b2)1/2.
– We use the notation | a+bi | to stand for the
absolute value a2+b2.
• Division: (1+2i)/(2 – i)
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The complex plane
Im
1+2i
3+i
Re
2–i
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Polar form
Im
a+bi = r (cos  + i sin ) = r ei
a
r

Re
b
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COMPLEX MATRICES
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Complex vectors and matrices
• Complex vector: vector with complex entries
– Examples:
• Complex matrix: matrix with complex entries
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Length of complex vector
• If we apply the calculation of the length of a
vector to a complex, something strange may
happen.
– Example: the “length” of [i 1] would be
– Example: the “length” of [2i 1] would be
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Definition
• The norm, or length, of a complex vector
[z1 z2 … zn]
where z1, z2, … zn are complex numbers, is
defined as
• Example
– The norm of [i 1] is
– The norm of [2i 1] is
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Complex dot product
• For complex vector, the dot product
is replaced by
where c1, d1, e1, c2, d2, e2 are complex numbers
and c1*, d1*, and e1* are the conjugates of c1,
d1, and e1 respectively.
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The Hermitian operator
• The transpose operator for real matrix should
be replaced by the Hermitian operator.
• The conjugate of a vector v is obtained by
taking the conjugate of each component in v.
• The conjugate of a matrix M is obtained by
taking the conjugate of each entry in M.
• The Hermitian of a complex matrix M, is
defined as the conjugate transpose of M.
• The Hermitian of M is denoted by MH or .
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Example
Hermitian
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Example
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Complex matrix in special form
•
•
•
•
Hermitian: AH=A.
Skew-Hermitian: AH= –A.
Unitary: AH =A-1, or equivalently AH A = I.
Example:
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• Dec 24, 1822 – Jan 14, 1901.
• French mathematician
• Introduced the notion of
Hermitian operator
• Proved that the base of the
natural log, e, is transcendental.
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http://en.wikipedia.org/wiki/Charles_Hermite
Charles Hermite
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Properties of Hermitian matrix
Let M be an nn complex Hermitian matrix.
• The eigenvalues of M are real numbers.
• We can choose n orthonormal eigenvectors of M.
– n vectors v1, v2, …, vn, are called “orthonormal” if they
are (i) mutually orthogonal viH vj =0 for i j, and
(ii) viH vi =1 for all i.
• We can find a unitary matrix U, such that M can
be written as UDUH, for some diagonal matrix
with real diagonal entries.
http://en.wikipedia.org/wiki/Hermitian_matrix
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Properties of skew-Hermitian
matrix
Let S be an nn complex skew-Hermitian matrix.
• The eigenvalues of S are purely imaginary.
• We can choose n orthonormal eigenvectors of
S.
• We can find a unitary matrix U, such that S
can be written as UDUH, for some diagonal
matrix with purely imaginary diagonal entries.
http://en.wikipedia.org/wiki/Skew-Hermitian_matrix
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Properties of unitary matrix
Let U be an nn complex unitary matrix.
• The eigenvalues of U have absolute value 1.
• We can choose n orthonormal eigenvectors of
U.
• We can find a unitary matrix V, such that U
can be written as VDVH, for some diagonal
matrix whose diagonal entries lie on the unit
circle in the complex plane.
http://en.wikipedia.org/wiki/Unitary_matrix
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Eigenvalues of Hermitian, skewHermitian and unitary matrices
Im
Complex plane
Hermitian
Skew-Hermitian
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Generalization: Normal matrix
A complex matrix N is called normal, if
NH N = N N H .
• Normal matrices contain symmetric, skewsymmetric, orthogonal, Hermitian, skewHermitain and unitary as special cases.
• We can find a unitary matrix U, such that N
can be written as UDUH, for some diagonal
matrix whose diagonal entries are the
eigenvalues of N.
http://en.wikipedia.org/wiki/Normal_matrix
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COMPLEX EXPONENTIAL FUNCTION
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Exponential function
• Definition for real x:
y = ex.
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y
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3
2
1
0
-2
-1.5
-1
-0.5
0
x
0.5
1
1.5
2
http://en.wikipedia.org/wiki/Exponential_function
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Derivative of exp(x)
y= ex
3
2
1
y
For example, the slope
of the tangent line at
x=0 is equal to e0=1.
0
-1
-2
-3
-3
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Taylor series expansion
• We extend the definition of exponential
function to complex number via this Taylor
series expansion.
• For complex number z, ez is defined by simply
replacing the real number x by complex
number z:
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http://en.wikipedia.org/wiki/Taylor_series
Series expansion of sin and cos
• Likewise, we extend the definition of sin and
cos to complex number, by simply replacing
real number x by complex number z.
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Example
• For real number :
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Euler’s formula
For real number ,
Proof:
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Summary
• Matrix and vector are extended from real to
complex
– Transpose  conjugate transpose (Hermitian
operator)
– Symmetric  Hermitian
– Skew-symmetric  skew-Hermitian
• Exponential function and sinusoidal function
are extended from real to complex by power
series.
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