Vai's Uni Notes

Cayley-Hamilton Theorem

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Every matrix satisfies its own characteristic equation; consequently every power series in 𝐴 collapses to a polynomial of degree 𝑛−1.

Statement

Cayley-Hamilton Theorem

  • Every matrix satisfies its own char. eq’n:
  • 𝑝𝐴(𝐴)=0𝑛×𝑛
  • i.e.
    • 𝐴𝑛+𝛼𝑛−1𝐴𝑛−1+…+𝛼1𝐴+𝛼0𝐼=0𝑛×𝑛
    • where 𝐴2=𝐴⋅𝐴, etc.
    • and 𝛼0,𝛼1,…,𝛼𝑛−1 are the coefs. in the char. poly that defines the eigenvalues
  • Equivalently:
    • 𝐴𝑛=−[𝛼0𝐼+𝛼1𝐴+…+𝛼𝑛−1𝐴𝑛−1]
    • i.e. 𝐴𝑛 is a linear combination of 𝐴0=𝐼 up to 𝐴𝑛−1

Higher Powers of A

  • Hence
    • 𝐴𝑛+1=−[𝛼0𝐴+𝛼1𝐴2+…+𝛼𝑛−1𝐴𝑛]
    • can also be written as a linear combination of 𝐴0,…,𝐴𝑛−1 (substitute for 𝐴𝑛 from above)
  • By induction we have:
    • For any 𝑘≥𝑛, 𝐴𝑘 can be written as a linear combination of 𝐴𝑗, 0≤𝑗≤𝑛−1

Finite Series for the Matrix Exponential

  • Recall: for every 𝑘≥𝑛, 𝐴𝑘 can be written as a linear combination of 𝐴0=𝐼,𝐴,𝐴2,…,𝐴𝑛−1
  • This applies to each term in the ∞ series for the matrix exponential 𝑒𝐴𝑡, so that
    • ∑∞𝑘=0𝑡𝑘𝑘!𝐴𝑘⏟∞series=∑𝑛−1𝑘=0𝛼𝑘(𝑡)𝐴𝑘⏟finite series(𝑛terms)
  • The 𝛼𝑘(𝑡) can be found from
    • [11⋮1𝜆1𝜆2⋮𝜆𝑛𝜆21𝜆22⋮𝜆2𝑛………𝜆𝑛−11𝜆𝑛−12⋮𝜆𝑛−1𝑛]⏟Vandermonde matrix[𝛼0(𝑡)𝛼1(𝑡)⋮𝛼𝑛−1(𝑡)]=[𝑒𝜆1𝑡𝑒𝜆2𝑡⋮𝑒𝜆𝑛𝑡]
  • ⇒ each 𝛼𝑘(𝑡) is a linear combination of 𝑒𝜆𝑘𝑡 terms, 𝜆𝑘 eigenvalues of 𝐴
  • This way of seeing that 𝑒𝐴𝑡 is built from the 𝑒𝜆𝑘𝑡 is less useful computationally, but quite helpful in certain theoretical arguments (compare the Laplace approach)

Possible slip in notes

  • The handwritten Vandermonde system labels the unknowns 𝛼1(𝑡),…,𝛼𝑛(𝑡) (and the first entry on the right as 𝑒𝜆𝑡); written here with the indices 0,…,𝑛−1 to match the finite sum

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