ENGG2013 Unit 20 Extensions to Complex numbers

1. ENGG2013 Unit 20Extensions to Complex numbers Mar, 2011.

2. 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]. ENGG2013

3. Examples • We usually denote the norm of a vector v by || v ||. ENGG2013

4. 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 ENGG2013

5. REVIEW OF COMPLEX NUMBERS ENGG2013

6. Quadratic equation • When the discriminant of a quadratic equation is negative, there is no real solution. • The complex rootsare ENGG2013

7. Complex eigenvalues • There are some matrices whose eigenvalues are complex numbers. • The characteristic polynomial of this matrix is The eigenvalues are ENGG2013

8. 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. ENGG2013

9. 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) ENGG2013

10. The complex plane Im 1+2i 3+i Re 2 – i ENGG2013

11. Polar form Im a+bi = r (cos  + i sin ) = r ei a r  Re b ENGG2013

12. COMPLEX MATRICES ENGG2013

13. Complex vectors and matrices • Complex vector: vector with complex entries • Examples: • Complex matrix: matrix with complex entries ENGG2013

14. 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 ENGG2013

15. 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 ENGG2013

16. 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. ENGG2013

17. The Hermitian operator • The transpose operator for real matrix should be replaced by the Hermitian operator. • The conjugate of a vectorv 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 . ENGG2013

18. Example Hermitian ENGG2013

19. Example ENGG2013

20. Complex matrix in special form • Hermitian: AH=A. • Skew-Hermitian: AH= –A. • Unitary: AH =A-1, or equivalently AH A= I. • Example: ENGG2013

21. Charles Hermite http://en.wikipedia.org/wiki/Charles_Hermite • Dec 24, 1822 – Jan 14, 1901. • French mathematician • Introduced the notion ofHermitian operator • Proved that the base of thenatural log, e, is transcendental. ENGG2013

22. 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 ENGG2013

23. 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 ENGG2013

24. 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 ENGG2013

25. Eigenvalues of Hermitian, skew-Hermitian and unitary matrices Im Complex plane unitary Hermitian Re Skew-Hermitian 1 ENGG2013

26. 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. http://en.wikipedia.org/wiki/Normal_matrix ENGG2013

27. COMPLEX EXPONENTIAL FUNCTION ENGG2013

28. Exponential function • Definition for real x: y = ex. http://en.wikipedia.org/wiki/Exponential_function ENGG2013

29. Derivative of exp(x) y= ex y=1+x For example, the slopeof the tangent line atx=0 is equal to e0=1. ENGG2013

30. 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: ENGG2013

31. 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. http://en.wikipedia.org/wiki/Taylor_series ENGG2013

32. Example • For real number : ENGG2013

33. Euler’s formula For real number , Proof: ENGG2013

34. 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. ENGG2013