Vai's Uni Notes

Dyadic Expansion

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Writing the matrix exponential as a sum of rank-one “dyads” built from right and left eigenvectors (distinct eigenvalues assumed; see diagonalization).

Matrix Exponential by Diagonalization

  • How to exploit 𝐴=𝑉Λ𝑉−1? Note that, for any 𝑘:
    • 𝐴𝑘=(𝑉Λ𝑉−1)𝑘=(𝑉Λ𝑉−1)(𝑉Λ𝑉−1)(𝑉Λ𝑉−1)…(𝑉Λ𝑉−1)⏟𝑘times
    • so 𝐴𝑘=𝑉Λ𝑘𝑉−1
    • when 𝑘=0: 𝐴0=𝐼=𝑉𝑉−1
  • Thus
    • 𝑒𝐴𝑡=∑∞𝑘=0𝑡𝑘𝑘!𝐴𝑘⇒𝑒𝐴𝑡=𝑉[∑∞𝑘=0𝑡𝑘𝑘!Λ𝑘]⏟𝑒Λ𝑡=diag{𝑒𝜆𝑘𝑡}𝑉−1

Summary (so far)

  • 𝑒𝐴𝑡=𝑉𝑒Λ𝑡𝑉−1
  • 𝑉: eigenvector matrix for 𝐴
  • 𝑒Λ𝑡=diag{𝑒𝜆𝑘𝑡}, 𝜆𝑘 eigenvalues

Dyadic Expansion

  • Now let
    • 𝑉−1=[𝐰𝖳1𝐰𝖳2⋮𝐰𝖳𝑛](row partition)
    • i.e. 𝐰𝖳𝑘 = 𝑘th row of 𝑉−1 (these are the left eigenvectors)
  • so that
    • 𝑒𝐴𝑡=[𝐯1,𝐯2,…,𝐯𝑛](𝑒𝜆1𝑡𝑒𝜆2𝑡⋱𝑒𝜆𝑛𝑡)[𝐰𝖳1𝐰𝖳2⋮𝐰𝖳𝑛]=[𝐯1,𝐯2,…,𝐯𝑛][𝑒𝜆1𝑡𝐰𝖳1𝑒𝜆2𝑡𝐰𝖳2⋮𝑒𝜆𝑛𝑡𝐰𝖳𝑛]=𝑒𝜆1𝑡𝐯1𝐰𝖳1+…+𝑒𝜆𝑛𝑡𝐯𝑛𝐰𝖳𝑛

Dyadic Expansion of the Matrix Exponential

  • Thus
  • 𝑒𝐴𝑡=∑𝑛𝑘=1𝑒𝜆𝑘𝑡𝐯𝑘𝐰𝖳𝑘
  • “Dyadic expansion” of 𝑒𝐴𝑡
  • Each 𝐯𝑘𝐰𝖳𝑘 is a “dyad” (expands to an 𝑛×𝑛 matrix after multiplication)
  • What are these 𝐰𝖳𝑘 terms (beyond the purely technical definition above)? → they are the left eigenvectors of 𝐴, bi-orthonormal to the 𝐯𝑘 (see Bi-Orthonormality)

Example

Example Application: Calculate the Matrix Exponential

  • 𝐴=[0−61−5]
  • 𝜆1=−2𝜆2=−3𝐯1=[1−2]𝐯2=[−13]
  • Λ=[−200−3]𝑉=[1−2−13]
  • Calculate
    • 𝑉−1=[3211]←rows:𝐰𝖳1=[3,1],𝐰𝖳2=[2,1]
    • 𝐯1𝐰𝖳1=[3−61−2]𝐯2𝐰𝖳2=[−26−13]
  • 𝑒𝐴𝑡=𝑒−2𝑡[3−61−2]+𝑒−3𝑡[−26−13]
  • (Matches the Laplace / partial-fraction result)

Application: Homogeneous Response

  • Applying the dyadic expansion to ̇𝐱=𝐴𝐱 gives the modal decomposition of the response:
    • 𝐱(𝑡)=∑𝑛𝑘=1(𝐰𝖳𝑘𝐱0)𝑒𝜆𝑘𝑡𝐯𝑘

Graph View

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