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].kshum ENGG2013 2
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.
• The complex rootsare
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-2 -1 0 1 2 3 40
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x
y
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
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Re
Im
1+2i
2 – i
3+i
Polar form
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Re
Im
a+bi = r (cos + i sin ) = r ei
r
a
b
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 .kshum ENGG2013 17
Example
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Hermitian
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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Charles Hermite
• 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/w
iki/C
harles_Herm
ite
Properties of Hermitian matrix
Let M be an nn 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 vi
H vj =0 for i j, and (ii) vi
H 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.
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http://en.wikipedia.org/wiki/Hermitian_matrix
Properties of skew-Hermitian matrix
Let S be an nn 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.
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http://en.wikipedia.org/wiki/Skew-Hermitian_matrix
Properties of unitary matrix
Let U be an nn 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.
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http://en.wikipedia.org/wiki/Unitary_matrix
Eigenvalues of Hermitian, skew-Hermitian and unitary matrices
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HermitianRe
Im Complex plane
1S
kew-
Herm
itian
unitary
Generalization: Normal matrix
A complex matrix N is called normal, if NH N = N NH.• Normal matrices contain symmetric, skew-
symmetric, orthogonal, Hermitian, skew-Hermitain 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.
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http://en.wikipedia.org/wiki/Normal_matrix
COMPLEX EXPONENTIAL FUNCTION
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Exponential function
• Definition for real x:
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x
y
y = ex.
http://en.wikipedia.org/wiki/Exponential_function
Derivative of exp(x)
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-3 -2 -1 0 1 2 3-3
-2
-1
0
1
2
3
x
y
y= ex
y=1+
x
For example, the slopeof the tangent line atx=0 is equal to e0=1.
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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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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http
://e
n.w
ikip
edia
.org
/wik
i/Tay
lor_
serie
s
Example
• For real number :
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Euler’s formula
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For real number ,
Proof:
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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