Puissances de matrices

8.1 Definition

Pour une matrice carree A :

A⁰ = I   (matrice identite)
A¹ = A
A² = A x A
A³ = A x A x A = A² x A
Aⁿ = A x A x ... x A   (n fois)

A = [ 1  2 ]
    [ 0  3 ]

A² = A x A :

c₁₁ = 1 x 1 + 2 x 0 = 1 + 0 = 1
c₁₂ = 1 x 2 + 2 x 3 = 2 + 6 = 8
c₂₁ = 0 x 1 + 3 x 0 = 0 + 0 = 0
c₂₂ = 0 x 2 + 3 x 3 = 0 + 9 = 9

A² = [ 1  8 ]
     [ 0  9 ]

A³ = A² x A :

c₁₁ = 1 x 1 + 8 x 0 = 1 + 0 = 1
c₁₂ = 1 x 2 + 8 x 3 = 2 + 24 = 26
c₂₁ = 0 x 1 + 9 x 0 = 0 + 0 = 0
c₂₂ = 0 x 2 + 9 x 3 = 0 + 27 = 27

A³ = [ 1  26 ]
     [ 0  27 ]

Si D est diagonale, alors Dⁿ est obtenue en elevant chaque element diagonal a la puissance n.

D = [ 2  0 ]      D³ = [ 2³  0  ] = [ 8   0 ]
    [ 0  5 ]           [ 0   5³ ]   [ 0  125 ]