Diagonalization II
If a matrix is orthogonally diagonalizable, what two things must also be true?
1) There is an orthonormal basis of Rn consisting of n eigenvectors
2) It is a symmetric matrix
Define the two parts of a complex number z = a + ib
a is the real part of z; Re(z)
b is the imaginary part of z; Im(z)
What is zz̄ and z-1 equal to?
1) |z|2
2) z̄/|z|2
State Euler's Formula and what four values it cycles through.
eit = cos(t) + isin(t)
i → -1 → -i → 1
How do you mathematically multiply two complex numbers?
Multiply their norms together and add their arguments
If A is a symmetric matrix, are all eigenvalues of A real numbers?
Yes
How do you add complex numbers?
Add real with real, imaginary with imaginary
For two complex numbers w and z, how do you compute w/z?
w/z = wz-1= wz̄/|z|2
What is each of the following equal to?
1) Re(eit)
2) Im(eit)
3) e-it
4) |e-it|
5) (e-it)-1
1) cos(t)
2) sin(t)
3) its complex conjugate
4) 1 for all t ∈ R
5) e-it
How do you compute a complex number to some power?
Raise its norm to that same power and multiply its argument by that power
If A is a symmetric matrix, are all eigenvectors from different eigenspaces of A automatically orthogonal?
Yes
How do you multiply two complex numbers?
FOIL and separate real from imaginary
What are the projection formulas for Re(z) and Im(z)?
Re(z) = (z + z̄)/2
Im(z) = (z - z̄)/2i
What set lies within the unit circle in the complex plane?
z ∈ ℂ and |z| = 1 ⇒ |z - 0| = 1 ⇒ d(z, 0) = 1 ⇒ z is on the unit circle
What does a scaling-rotation matrix look like?
[[a -b], [b a]]
How do you find a symmetric matrix with real number entries?
1) Eigenvalues
2) Eigenspaces
3) Graham Cracker
4) Each orthonormal basis vector is a column for the matrix
What is the complex conjugate of z = a + ib
z̄ = a - ib (Think of it as a reflection about the real axis)
T/F:
1) Conjugation do not respect all arithmetic operations
2) Norms are multiplicative
1) F
2) T
What is the polar/exponential form of a complex number?
BONUS 100 POINTS: What does θ represent?
z = |z|eiθ
Bonus: θ = arg(z) which represents the angle between the positive Re(z)-axis and the line joining 0 to z
For some scaling-rotation matrix of the form [[a -b], [b a]], what do the scaling and rotation portions look like?
Scaling: [[r 0], [0 r]] for r = |a + ib|
Rotation: [[cosθ -sinθ], [sinθ cosθ]] for tanθ = b/a
When is P-1AP = D an orthogonally diagonalizable?
When P-1 = PT
What is the modulus of z?
|z| = √a2 + b2 (Magnitude of z)
How do you find the distance between two complex numbers, z and w?
d(z, w) = |z - w|
What is the geometry behind multiplying two complex numbers?
Scaled rotation;
The amount of rescaling is |z|
the amount of rotation is θ = arg(z)
For some scaling-rotation matrix, what are its eigenvalues?
a + ib and a - ib