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ENGG2013 Unit 20 Extensions to Complex numbers Mar, 2011.

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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, [a 1 a 2 … a n ], is defined as, which is also called the norm of [a 1 a 2 … a n ]. kshumENGG20132

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Examples We usually denote the norm of a vector v by || v ||. kshumENGG20133

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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 kshumENGG20134

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REVIEW OF COMPLEX NUMBERS kshumENGG20135

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Quadratic equation When the discriminant of a quadratic equation is negative, there is no real solution. The complex roots are kshumENGG20136

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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 kshumENGG20137

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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–2i 2 =4+3i. kshumENGG20138

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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) = (a 2 +b 2 ) 1/2. – We use the notation | a+bi | to stand for the absolute value a 2 +b 2. Division: (1+2i)/(2 – i) kshumENGG20139

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The complex plane kshumENGG Re Im 1+2i 2 – i 3+i

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Polar form kshumENGG Re Im a+bi = r (cos + i sin ) = r e i r a b

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COMPLEX MATRICES kshumENGG201312

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Complex vectors and matrices Complex vector: vector with complex entries – Examples: Complex matrix: matrix with complex entries kshumENGG201313

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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 kshumENGG201314

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Definition The norm, or length, of a complex vector [z 1 z 2 … z n ] where z 1, z 2, … z n are complex numbers, is defined as Example – The norm of [i 1] is – The norm of [2i 1] is kshumENGG201315

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Complex dot product For complex vector, the dot product is replaced by where c 1, d 1, e 1, c 2, d 2, e 2 are complex numbers and c 1 *, d 1 *, and e 1 * are the conjugates of c 1, d 1, and e 1 respectively. kshumENGG201316

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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 M H or. kshumENGG201317

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Example kshumENGG Hermitian

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Example kshumENGG201319

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Complex matrix in special form Hermitian: A H =A. Skew-Hermitian: A H = –A. Unitary: A H =A -1, or equivalently A H A = I. Example: kshumENGG201320

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Charles Hermite Dec 24, 1822 – Jan 14, French mathematician Introduced the notion of Hermitian operator Proved that the base of the natural log, e, is transcendental. kshumENGG

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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 v 1, v 2, …, v n, are called orthonormal if they are (i) mutually orthogonal v i H v j =0 for i j, and (ii) v i H v i =1 for all i. We can find a unitary matrix U, such that M can be written as UDU H, for some diagonal matrix with real diagonal entries. kshumENGG

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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 UDU H, for some diagonal matrix with purely imaginary diagonal entries. kshumENGG

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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 VDV H, for some diagonal matrix whose diagonal entries lie on the unit circle in the complex plane. kshumENGG

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Eigenvalues of Hermitian, skew- Hermitian and unitary matrices kshumENGG Hermitian Re Im Complex plane 1 Skew-Hermitian unitary

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Generalization: Normal matrix A complex matrix N is called normal, if N H N = N N H. 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 UDU H, for some diagonal matrix whose diagonal entries are the eigenvalues of N. kshumENGG

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COMPLEX EXPONENTIAL FUNCTION kshumENGG201327

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Exponential function Definition for real x: kshumENGG y = e x.

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Derivative of exp(x) kshumENGG y= e x y=1+x For example, the slope of the tangent line at x=0 is equal to e 0 =1.

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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, e z is defined by simply replacing the real number x by complex number z: kshumENGG201330

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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. kshumENGG

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Example For real number : kshumENGG201332

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Eulers formula kshumENGG 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. kshumENGG201334

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