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Matrice orthogonale

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Résolution de matrice orthogonal Les valeurs propres. orthogonale.

Institution
Aix-Marseille (AMU)
Course
Mathématique pour l\\\'ingénieur








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Aix-Marseille (AMU)
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génie civil
Course
Mathématique pour l\\\'ingénieur
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Uploaded on
September 4, 2025
Number of pages
1
Written in
2022/2023
Type
Class notes
Professor(s)
M burtchelle
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Subjects

  • matrice
  • orthogonale

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Matrice orthogonale
jeudi 20 octobre 2022 08:18



A est symétrique si A=AT
E, espace euclidien rapporté à une base orthonormée

X . f(y) = y . f(x)
==> f est un opérateur symétrique




XTAY = YTAX ??

XTAY = (XTAY)T
= (AY)TX
=YTATX
=YTAX

• Les valeurs propres d'une matrice symétrique (réelles) sont toutes réelles.
• Deux vecteurs propres associés à 2 valeurs propres distincte sont
orthogonaux.
• Toute matrice symétrique A d'ordre n est diagonalisable. Les valeurs
propres sont réelles et les vecteurs propres forment une base
orthogonale.
• Si les vecteurs propres sont choisis unitaires alors la base de vecteurs
propre est orthonormée et P orthogonale.

Matrice orthogonale : P-1 = PT

Det ( PTP) = det (PT)det (P)
D'une part
det (PT)det (P)= Det (p-1) det (P) = 1/det(P) det (P)
d'autre part
det (PT)det (P)= det ( P) det (P) = det (P²)

Det (P) = 1





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