Definitions

• eigenvalue of A
• real/complex number l
• Ax = lx
• nxn matrix A has n eigenvalues
• not necessarily all distinct
  
• eigenvector of A
• x in the above equation corresponding to each eigenvalue
• not unique, any scalar multiple works
• Calculating eigenvalues/vectors
• (A-lI)x = 0
• homogenous equation, need to find nonzero solutions
• find l such that A-lI is singular
• Characteristic Equation
• det(A-lI) - monic polynomial of degree n
  
• A-lI is singular <--> determinant is 0
• roots of characteristic equation are the eigenvalues of A
• companion form matrix can be formed using the coefficients
• TODO

Distinct eigenvalues

• n eigenvalues -> n linearly independent eigenvectors
• Aqi = liqi
• {q1 q2 ... qn} form a basis
• diagonal matrix representation
• A_bar = diagonal matrix with eigenvalues as diagonal values
• A_bar = Q^-1AQ
• easily checked using QA_bar = AQ

Repeated Eigenvalues

• repeated eigenvalues of A
• eigenvalues with multiplicity higher than 1
• 1 = simple eigenvalue
• results in block-diagonal & triangular-form representation
• nxn matrix with 1 eigenvalue of multiplicity n
• solutions to (A-lI)q = 0
• #independent solutions = nullity of (A-lI)
• need (n - #independent solutions above) more solutions to form basis
• provided by generalized eigenvectors
• Generalized eigenvectors
• of grade n = v
• (A-lI)ⁿv = 0
• (A-lI)^n-1v != 0
• chain of generalized eigenvectors
• for each independent solution to (A-lI) = 0
• obtain needed eigenvectors of increasing grade
• eg: for n = 4, nullity = 1
• eigenvectors {v1...v4}
• (A-lI)v₁ = 0
• (A-lI)²v₂ = 0
• (A-lI)³v₃ = 0
• (A-lI)⁴v₄ = 0
• Representation of A w.r.t basis {TODO}
• jordan block of order n
• TODO
• for 1 eigenvalue and nullity = 1
• eigenvalue on diagonal and 1 on superdiagonal 
• due to one chain of generalized eigenvectors
•  for 1 eigenvalue and nullity = 2
• two linearly independent eigenvectors
• need (n-2) generalized eigenvectors
• two chains of generalized eigenvectors
• {v1...vk,u1...um} and k+m = n
• v and u are the two linearly independent soln.
• two jordan blocks of order k and m
• A = diag{J1,J2}
• for 2 eigenvalues, l1 repeated and l2 simple
• l1 nullity = 1
• one jordan block with l2 on the 5th diagonal cell 
• l1 nullity = 2
• two jordan blocks assosiated with l1
• l1 nullity = 3
• three jordan blocks (with two of them degenerated)
• l1 nullity = 4
• four degenerated jordan blocks
• degenerated jordan blocks
• jordan blocks with order 1
• defective matrix
• a matrix that cannot be diagonalized
• instead, it's converted to jordan form

Properties

• det(CD) = det(C)det(D)
• det(Q)det(Q^-1) = det(I) = 1
• det(A) = det(A_jordan)
• det(A_jordan) = product of all eigenvalues
• A is nonsingular IFF it has no 0 eigenvalue
• nilpotent matrix J
• (J-lI)ⁿ = 0 for n>=k, some k

Functions of Square Matrix

• powers of matrix A
• A^k = A*A*... n times
• A^0 = I
• A^k = Q^-1A_j^kQ
• polynomial function of matix
• f(x) for some scalar x
• polynomials we've seen in high school
• f(A) for some matrix A
• same polynomial as f(x) w/ x replaced by A
• f(A) for some block diagonal A = [A1 0; 0 A₂] (A1A2 are square matrix blocks)
• A^k = [A₁^k 0; 0 A₂^k 0]
• f(A) = [f(A1) 0; 0 f(A2)]
• function in terms of jordan form
• f(A) = Q^-1f(Aj)Q
• monic polynomial
• 1 as the leading coefficient
• minimal polynomial of A
• psi(l)
• monic polynomial of least degree such that psi(A) = 0 (nxn 0 matrix)
• A and its jordan form have same minimal polynomial
• calculation
• TODO
• equal to characteristic polynomial if
• all eigenvalues are distinct
• psi(A) = 0

Cayley-Hamilton Theorem

• characteristic polynomial of A is satisfied by A
• c(l) = det(lI-A) --> c(A) = det(AI-A) = 0
•  Minimal polynomial of A is satisfied by A
• psi(l) is a factor of c(l), therefore psi(A) = 0 -> c(l) = 0
• Aⁿ and higher powers of A can be written as linear combination of
• {I, A, ..., A^n-1}
• polynomial function f(A) of any degree
• can be expressed as linear combination of {I, A, ..., A^n-1}
• f(A) = bI + b₁A + ... + b_n-1A^n-1
• can be expressed even smaller as lin. comb. of {I, A, ..., A^m-1}
• m is degree of minimal polynomial of A
• computing h(A) = bI + b₁A + ... + b_n-1A^n-1 for f(A)
• spectrum of A
• h(l) and f(l) are equal on spectrum of A if f(A) = h(A)
• higher derivatives of l are equal as well
• use long division
• f(l) = quotient(l)*det(l) + reminder(l)
• f(A) = 0 + reminder(A) = h(A)
  
• use spectrum properties of A to get n equations for b's
• f(l) = quotient(l)*det(l) + reminder(l)
  
• f(l) = reminder(l) = h(l) when l = eigenvalue of A
• also works for higher derivatives of l
  
• obtain m equations for each of the m eigenvalues of A
• for repeated roots of order p
• obtain p equations by taking ith derivative of f and h, and plugging in eigenvalue
  
• use n equations to solve for b and compute h(A)

Functions of a Square Matrix

• f(l) can be any function, not just polynomials
• f(l) = h(l) on the spectrum of A
• h(l) is a polynomial of degree n-1
• n = order of A
• examples
• A^100
• f(l) = l^100
• exp(At)
• when deriving equations for repeated roots, differentiation is w.r.t. l (not t)
• exp(At) for A = jordan block of order 4
• h(l) = b0 + b1(l - l₁) + b2(l - l₁)² + b3(l - l₁)³
  
• allows for easy calculation of b's
  
• h(A) = b + b1(A - l₁I) + b2(A - l₁I)² + b3(A - l₁I)³
  
• bi = f⁽ⁱ⁾(l₁)/i!
  
• (A-lI)^n have special properties (refer nilpotent matrices)
  
• exp(At) for A = block diagonal with 2 jordan blocks
• calculate each jordan block independently
• (s-l)^-1 for A above

Power Series