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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2

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10

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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-2 -1.5 -1 -0.5 0 0.5 1 1.5 20

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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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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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