21 - Eigen Values and Eigen Vectors, det(A - λI) = 0, Trace = λ₁ + λ₂ + … + λ_n

• x is an Eigen vector if **Ax

  x**

• λ is an Eigen value if Ax = λx
• If A is singular, λ = 0 is an eigen value
• Set of eigen values - spectrum

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• What are the eigen vectors and values for Projection matrix?

View Answer

> Eigen vectors - All vectors in projection plane, Eigen value - 1 > Eigen vectors - All vectors perpendicular to projection plane, Eigen value - 0

• What are the eigen vectors and values for [[0 1] [1 0]]?
  • Hint: Recall visualizing eigen vectors and values from 3b1b video (vectors which only scales up or down after transforming the space by matrix)

View Answer

> Seeing the matrix it seems the space has flipped around X and Y axis, hence lines at 45^o will be at same orientation > Eigen vectors - [1 1], [1 -1], Eigen value - 1, -1

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• Sum(λ) = Trace, Product(λ) = Determinant
• Solving Ax = λx is same as solving (A - λI)x = 0
  • (A - λI) is singular, since x != 0 :thinking:

  • Hence we solve

    A - λI

    = 0

  • This is known as characteristic equation

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• What are the eigen vectors and values for Rotation matrix R_90^o?

View Answer

> Rotation matrix is [[0 -1] [1 0]] > No eigen values or vectors

• if Ax = λx, find eigen values for (A + 2I)

View Answer

> (A + 2I)x = Ax + 2Ix = λx + 2x = (λ + 2)x

• if A and B have eigen values λ and α respectively, find eigen values for (A + B)

View Answer

> (λ + α) ? :grin: > No, since eigen vectors of A and B can be different

22 - Diagonalizing a matrix (S_-1AS = Λ), Powers of A, u_k+1 = Au_k

• Suppose there are n independent eigen vectors of A
• Let S be matrix formed with columns containing eigen vectors
• AS = A[x1 x2 x3 … xn] = [λ1x1 λ2x2 …. λnxn] = [x1 x2 x3 …. xn]diagonal([λ1 λ2 … λn])
• AS = SΛ
• A = SΛS^-1 OR Λ = S^-1AS
• A^k = SΛ^kS^-1
• A is sure to have n independent eigen vectors and be diagonalizable if all λ’s are different
• Repeated eigen values may or may not result in n independent eigen vectors
  • eg. Identity matrix
• Algebraic multiplicity - Number of times an eigen value is repeated
• Geometric multiplicity - Number of eigen vectors for a particular eigen value

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• Fibonacci series - 0, 1, 1, 2, 3, 5, 8…. F₁₀₀ = ?
• F_{k + 2} = F_{k + 1} + F_k … We want to write this as u_k+1 = Au_k
• F_{k + 1} = F_{k + 1}
• Trick: Let u_k = [F_{k + 1} F_k]
• Your work:-
  • Find A, eigen value, vector
  • Use following logic for finding F₁₀₀
  • u₀ = c₁x₁ + c₂x₂ + … + c_nx_n = SC
  • A^ku₀ = Λ^kSC

23 - Differential equation (du/dt = Au), Exponent of matrix (e^At)

• du / dt = Au … u(t) = c₁e^λ1x₁ + c₂e^λ2x₂ + … + c_ne^λnx_n
• Stability, steady state or blow up based on λ
  • Stability - λ < 0
  • Steady state - one λ = 0 and other < 0
  • Blow up - if any λ > 0
  • TODO - Explain e^At

24 - Markov matrices, Steady state, Fourier series and Projections

• Steady state -
  • λ = 0 if exponents
  • λ = 1 if powers
• Markov matrix -
  • All elements >= 0
  • All columns add to 1
• Power of Markov matrix is another Markov matrix
• λ = 1 is an eigen value of Markov matrix and corresponding eigen vector has all its components > 0 => positive steady state
• All other eigen values are between (-1, 1)
• TODO: Explain example of Markov matrix by migration system model
• V = QC (v = q₁c₁ + q₂c₂ …) …. q = orthonormal basis
• C = Q^-1V = Q^TV …. hence can find all coefficients
• TODO: Explain Fourier series

25 - Symmetric matrices (Eigen values and vectors) and Positive definite matrices

• Real symmetric matrices - Real eigen values and eigen vectors are orthogonal
• Positive definite symmetric matrix -
  • All eigen values are positive
  • All pivots are positive
  • All sub-determinant positive