1. Extensions to Complex numbers1

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].
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3. Examples We usually denote the norm of a vector v by || v ||. kshum
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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
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5. REVIEW OF COMPLEX NUMBERS
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6. Quadratic equation
When the discriminant of a quadratic equation is negative, there is no real solution.
The complex roots are
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7. 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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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.
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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)
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10. The complex plane Im
1+2i
3+i Re
2 – i
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11. Polar form Im a+bi = r (cos  + i sin ) = r ei a r  Re b kshum
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12. COMPLEX MATRICES
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13. Complex vectors and matrices
Complex vector: vector with complex entries
Examples:
Complex matrix: matrix with complex entries
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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
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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
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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.
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17. 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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18. Example
Hermitian
\|\mathbf{v}\|^2 = \mathbf{v}^H\mathbf{v} = \begin{bmatrix}
1&
-4i&
2-i
\end{bmatrix}\begin{bmatrix}
1\\
4i\\
2+i
\end{bmatrix} = = 22
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19. Example \begin{bmatrix} 2 & 2i & 1-i \\ -2i &0 & i\\ 1+i & -i & 1
\end{bmatrix}^H =\begin{bmatrix}
\end{bmatrix}
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20. 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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21. Charles Hermite Dec 24, 1822 – Jan 14, 1901. French mathematician
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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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.
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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.
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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.
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25. Eigenvalues of Hermitian, skew-Hermitian and unitary matricesIm
Complex plane
unitary
Hermitian Re
Skew-Hermitian 1
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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.
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27. COMPLEX EXPONENTIAL FUNCTION
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28. Exponential function Definition for real x: y = ex.
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29. Derivative of exp(x) y= ex y=1+x
For example, the slope of the tangent line at x=0 is equal to e0=1.
e^x = \sum_{n=0}^\infty \frac{x^n}{n!} = 1+x+\frac{x^2}{2}+\frac{x^3}{3!}+ \ldots.
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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:
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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.
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32. Example For real number : \begin{align*}
cos(i\theta) &= 1-\frac{(i\theta)^2}{2!}+\frac{(i\theta)^4}{4!}- \frac{(i\theta)^6}{6!}+\ldots\\
&= 1+ \frac{\theta^2}{2!}+\frac{\theta^4}{4!}+\frac{\theta^6}{6!}+\ldots
\end{align*}
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33. Euler’s formula
For real number ,
Proof:
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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.
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