Satz von Cayley-Hamilton

Beispiel 9.3

A = {{1, 2, 2 + 3  I}, {3, 
          0, I}, {2 - I, I, 
          1}}

charpol = Det[A - λ IdentityMatrix[3]]

charpol = Expand[charpol]

MatrixPower[A, 2]

MatrixPower[A, 3]

charpol

-MatrixPower[A, 3] + 
      2 MatrixPower[A, 
          2] + (11 + 4 I) A - (12 - 10 I) 
        IdentityMatrix[3]

Anwendung des Satzes von Cayley-Hamilton zur Berechnung von Matrixpotzenzen

A = {{1, -1}, {1, 1}}

MatrixPower[A, 10]

CharacteristicPolynomial[A, λ]

matrixpower[A, 0] = IdentityMatrix[2];
matrixpower[A, 1] = A; 
    matrixpower[A, n_] := 
      matrixpower[A, n - 2] . (2  A - 2  IdentityMatrix[2])

matrixpower[A, 10]

MatrixPower[A, 100]

matrixpower[A, 100]

Eigenvektoren

A = {{1, 2, 3}, {4, 5, 6}, {7, 8, 9}}

Eigensystem[A]

Aufgabe 6.7 (Strampp)

A = {{1, 0, 2}, {3, 0, 0}, {0, 0, I}}

charpol = CharacteristicPolynomial[A, λ]

Factor[charpol]

Eigensystem[A]