Complex Matrices, Discrete Probability - Advanced Engineering Math - Tutorial Slides, Slides of Engineering Mathematics

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l atrices, Discrete Probability

l Definition Eigenvalues r iti n MH^ = M real

• er itian MH^ = - M purely imaginary it ry MH^ = M-^1 unit magnitude

t a er itian matrix, i.e. MH=M t an eigenvalue of M, v be a corresponding eigenvector, Mv=λv t to show that λ is real, i.e. λ*=λ.

t vH^ v, which is a scalar, in two ways. ( ) vH(λv)=λvHv ( ) ( Hv)Hv=( v)Hv=(λv)Hv=λvHv s v is a nonzero vector, vHv is positive. r vHv=λvHv, we get λ*=λ.

r f : er itian matrix has real eigenvalues. (Lecture content)

l Definition Eigenvalues r iti n MH^ = M real

• er itian MH^ = - M purely imaginary it ry MH^ = M-^1 unit magnitude

t a nitary matrix, i.e. MH=M-^1 t an eigenvalue of M, v be a corresponding eigenvector, Mv=λv t to show that |λ|= 1 , i.e. λ*λ=1.

t ( v)H( v), in two ways. ( ) ( v)=(λv)H(λv)=λ*λ(vHv) ( ) ( v)=(vH^ H)(Mv)=vHMHMv=(vHv)

( Hv) (vHv) s v is a nonzero vector, vHv is positive, finally we can get λ*λ=1.

r f : ll eigenvalues of a Unitary matrix have absolute value equal to 1

i itary atrix, its determinant has absolute value 1. |det A|= 1

r f :

t(I) = det (AA-^1 ) = det (AAH) = det (A) det (AH) (f ll s fro det (AB)= det (A) det (B), see tutorial 5 )

t ( H) det ((A)T) = det (A) = (det (A))* (t t r inant of the conjugate transpose of A equals the complex j te of the determinant of A)

l e: 1 = det (A) (det (A))* = |det (A)|.

r f :

| t | | (^) i λi | = ∏i |λi | = 1 , follows from example 1 (c).

r k algorith

B

C

The matrix A of this web graph is,

0 0 0 1 1 0 0 0 0 1 0 0 0 0 1 0

⎡ ⎤ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦

) angling node; [ ling node eans that it does not contain any out-going link]

) irr ucible graph; [ r h is called irreducible if for any pair of distinct nodes, we can t rt fro one of them, follow the links in the web graph and arrive at t t er node, and vice versa.]

0 1 0 0 0 0 1 0

⎡ ⎤ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦

iteration xA xB xC xD 0 0. 25 0. 25 0. 25 0. 25 1 0. 25 0. 25 0. 25 0. 25 2 0. 25 0. 25 0. 25 0. 25 ... ... ... ... ... n 0. 25 0. 25 0. 25 0. 25

iteration xA xB xC xD 0 0. 5 0. 5 0 0 1 0 0. 5 0. 5 0 2 0 0 0. 5 0. 5 3 0. 5 0 0 0. 5 4 0. 5 0. 5 0 0

hat if?

.. 05 0. 05 0. 85 .. 05 0. 05 0. 05 .. 85 0. 05 0. 05 .. 05 0. 85 0. 05

⎤ ⎥ ⎥ ⎥ ⎥ ⎦

iteration xA xB xC xD 0 0. 25 0. 25 0. 25 0. 25 1 0. 25 0. 25 0. 25 0. 25 2 0. 25 0. 25 0. 25 0. 25 ... ... ... ... ... n 0. 25 0. 25 0. 25 0. 25

iteration xA xB xC xD 0 0. 5 0. 5 0 0 1 0. 05 0. 45 0. 45 0. 05 2 0. 09 0. 09 0. 41 0. 41 ... ... ... ... ... n 0. 25 0. 25 0. 25 0. 25

hat if?

.. 05 0. 05 0. 85 .. 05 0. 05 0. 05 .. 85 0. 05 0. 05 .. 05 0. 85 0. 05

⎤ ⎥ ⎥ ⎥ ⎥ ⎦

iteration xA xB xC xD 0 0. 5 0. 5 0 0 1 0. 05 0. 45 0. 45 0. 05 2 0. 09 0. 09 0. 41 0. 41 ... ... ... ... ... [ ] i ( ) n 0. 25 0. 25 0. 25 0. 25

. 0. 5000 - 0. 0000 - 0. 5000 i - 0. 0000 + 0. 5000 i . - 0. 5000 - 0. 5000 - 0. 5000 . 0. 5000 0. 0000 + 0. 5000 i 0. 0000 - 0. 5000 i . - 0. 5000 0. 5000 - 0. 0000 i 0. 5000 + 0. 0000 i . 0 0 0 - 0. 8000 0 0 0 0. 0000 + 0. 8000 i 0 0 0 0. 0000 - 0. 8000 i

r i g k Balls into n Bins

) ll/ is, bin/Dis, at most one ball per bin:

t : There are exactly k bins that contains one ball, and the re aining n-k bins are empty.

l ti : e choose k bins from the original n bins with order.

l : 2 balls and 3 bins.

3 2 1 = 6 1

× ×

r i g k Balls into n Bins

) ll/ is, bin/Dis, allow more than one ball per bin:

l ti : or each ball, it has n different choices.

l : 2 balls and 3 bins.

2 3 = 9

r i g k Balls into n Bins

) ll/Indis, bin/Dis, allow more than one ball per bin:

l ti : here are n bins, we use n- 1 bars to represent. So there r - k places. e choose n- 1 places to put bars, the remaining l ith balls.

l : 2 balls and 3 bins.

4 3 = 6 2 1

× ×

se there is a round table, we want to arrange seats for 6 people. s ating arrangements are considered different only when the iti s of people are different relative to each other.

) any kinds of different arrangement;

) If ny doesn't want to sit next to Lucy and James, then how many rr ents left?

) t od a: 3 2( 3 !)= 36

t od b: 5 !- 2 2( 4 !)+ 2 *( 3 !)= 36