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What is a complex number?

Z=a +ib
A is the real part
B is the imaginary


What is the complex conjugate?

Z=a +ib
Then Z bar = a -ib

The complex conjugate of (Z1 +Z2) = the conjugate of Z1 + conjugate of Z2


What is polar form?

Z= re^(i theta)
r = |z| = sqrt(a^2 + b^2)
Theta = arctan(a/b)


What is Euler’s formula?

Cos theta + Isin theta = e^(I theta)

E^(I pi) = -1


What is De Moivres Theorem?

(Cos theta + isin theta) ^n = cos(n theta) + I sin (n theta)


What is the complex roots of unity?

Z^n =1, Z= e^i theta
For theta = 2pi k/n , k= 0, ......, n-1


What is orthogonality and the orthogonal complement?

Vector U dotted with vector V = 0 - right angle

Orthogonal complement
S^( upside down T) of A sub space contains all vectors orthogonal to S


Projections onto a line

Projection of vector b onto a is
(a . b)/ |a|^2 x vector A

For a matrix
a a transpose /|a|^2 x vector X


Projections onto a plane

X = X - X.n/ |n|^2 x n

1. Find n = , normal to the plane
2. Project x onto n
X.n =
|n|^2 =
X.n/|n|^2 x n =
3. Subtract this from X to get the projection onto the plane


What is the grand-Schmidt algorithm?

Start with vectors a, b, c .... all linearly independent
1. Define A =a
2. B = b - b.A/|A|^2 xA
3. C = c - c.A/|A|^2 xA - c.B/|B|^2 x B
C.B=0 C.A=0
4. Normalise
e1 =A/ sqrt(|A|^2), e2= B/sqrt(|B|^2), e3....


For 2x2
And 3x3

|a b|
|c d| = ad - bc

|a b c|
|d e f| = a |e f| -b|d f| +c |d e|
|g h I | |h I| |g I| |g h|

+ in all the corners and then alternative


Properties of determinants

- Det =0 'singular', A^-1 does not exist
- Det A is not equal to zero there is a unique solution for Ax=b
- Det In = 1
- If A has two rows the same Det A =0
- Det lamda A = lamda^2 Det A
- Det(A transpose) = Det A
- Det(AB) = Det(A) x Det(B)
- If A has a row of zeros DetA =0


Eigenvalues and Eigenvectors

Av = lamda V
v is the eigenvector and lamda is the eigenvalue
1. Calculate Det(A-lamda I)=0 to get values for lamda
2. Eigenvectors from (A- lamda I) v=0

product of lamda = Det (a)
sum of lamda = TR(a) - trace of a = sum of the diagonal elements of A


Properties of eigenvectors and Eigenvalues

1. Av = lama v. v is not equal to 0, if lambda = 0 v is an element of N(A) and if lama is not =0 v is an element of C(A).
2. Av1=lamda1v1, Av2=lamda v2. lamda 1 not = lamda 2 then v1 and v2 are linearly independent
3. lama 1 not= lamda 2 implies v1 and v2 are linearly independent
lamda 1 = lamda 2, there may of may not exists 2 linearly independent vectors v1 and v2


Diagonalise Matrix A

1. Find lama j and Vj - eigenvalues and eigenvectors
2. construct x = ( v1 v2 ....)
3. calculate x^-1 A x = diagonal matrix, diagonal matrix of Eigen value
If A is diagonalisable, x^-1 AX = diagonal matrix


Powers of the diagonal matrix

The diagonal matrix to the power of n, has the diagonal elements of the matrix to the power n.
A = X(diagonal matrix) X^-1
A^m = X(diagonal matrix)^m X^-1


Matrix exponentials

e^A exists for all matrices (nxn)

If diagonalisable
e^A = Xe^(diagonal matrix) X^-1
e^(diagonal matrix) is e^of the eigenvalues on the diagonal


Symmetric and Hermitian Matrices

For real matrices S is symmetric iff S transpose =S
For complex matrices, S if Hermitian if S transpose =S
where S transpose = (S*) transpose = (S transpose)^*
cross symbol means complex conjugate and transpose

Real, symmetric implies hermitian

Two key properties of hermitian matrices
1. eigenvalues are real
2. always diagonalisable, even for repeated eigenvalues