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SciJewel
SciJewel
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mt1 = {{Cos[ξ] Cos[ϕ] - Sin[ξ] Sin[ϕ] Sin[ψ], Cos[ψ] Sin[ϕ], Cos[ϕ] Sin[ξ] + Cos[ξ] Sin[ϕ] Sin[ψ]},
       {-Cos[ξ] Sin[ϕ] - Cos[ϕ] Sin[ξ] Sin[ψ], Cos[ϕ] Cos[ψ], -Sin[ξ] Sin[ϕ] + Cos[ξ] Cos[ϕ] Sin[ψ]},
       {-Cos[ψ] Sin[ξ], -Sin[ψ], Cos[ξ] Cos[ψ]}};

mt2 = {{a, b, c}, {b, d, e}, {c, e, h}};

$$mt1 = \pmatrix{a& b& c\\b& d& e\\c& e&h}$$

HereFor mt1$mt1^T.mt2.mt1$ = d, where mt1 is an orthogonal matrix and mt2, mt2 is a real symmetric matrix. In principle, the output of Transpose[mt1].mt2.mt1 should be and d is a diagonal matrix, but Mathematica cannot do with eigenvalues of mt2 on the diagonal. How to find mt1 such that it is orthogonal and contains three variables using Mathematica. Any suggestion is appreciated!

mt1 = {{Cos[ξ] Cos[ϕ] - Sin[ξ] Sin[ϕ] Sin[ψ], Cos[ψ] Sin[ϕ], Cos[ϕ] Sin[ξ] + Cos[ξ] Sin[ϕ] Sin[ψ]},
       {-Cos[ξ] Sin[ϕ] - Cos[ϕ] Sin[ξ] Sin[ψ], Cos[ϕ] Cos[ψ], -Sin[ξ] Sin[ϕ] + Cos[ξ] Cos[ϕ] Sin[ψ]},
       {-Cos[ψ] Sin[ξ], -Sin[ψ], Cos[ξ] Cos[ψ]}};

mt2 = {{a, b, c}, {b, d, e}, {c, e, h}};

Here mt1 is an orthogonal matrix and mt2 is a real symmetric matrix. In principle, the output of Transpose[mt1].mt2.mt1 should be a diagonal matrix, but Mathematica cannot do it. Any suggestion is appreciated!

$$mt1 = \pmatrix{a& b& c\\b& d& e\\c& e&h}$$

For $mt1^T.mt2.mt1$ = d, where mt1 is an orthogonal matrix, mt2 is a real symmetric matrix and d is a diagonal matrix with eigenvalues of mt2 on the diagonal. How to find mt1 such that it is orthogonal and contains three variables using Mathematica. Any suggestion is appreciated!

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march
march
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mt1 = {{Cos[\[Xi]]Cos[ξ] Cos[\[Phi]]Cos[ϕ] - Sin[\[Xi]]Sin[ξ] Sin[\[Phi]]Sin[ϕ] Sin[\[Psi]]Sin[ψ], 
    Cos[\[Psi]]Cos[ψ] Sin[\[Phi]]Sin[ϕ], 
  Cos[ϕ] Sin[ξ] Cos[\[Phi]]+ Sin[\[Xi]]Cos[ξ] +Sin[ϕ] Sin[ψ]},
     Cos[\[Xi]] Sin[\[Phi]] Sin[\[Psi]]}, {-Cos[\[Xi]]Cos[ξ] Sin[\[Phi]]Sin[ϕ] - 
     Cos[\[Phi]]Cos[ϕ] Sin[\[Xi]]Sin[ξ] Sin[\[Psi]]Sin[ψ], 
    Cos[\[Phi]]Cos[ϕ] Cos[\[Psi]]Cos[ψ], -Sin[\[Xi]]Sin[ξ] Sin[\[Phi]]Sin[ϕ] + 
 Cos[ξ] Cos[ϕ] Sin[ψ]},
   Cos[\[Xi]] Cos[\[Phi]] Sin[\[Psi]]},  {-Cos[\[Psi]]Cos[ψ] Sin[\[Xi]]Sin[ξ], \
-Sin[\[Psi]]Sin[ψ], Cos[\[Xi]]Cos[ξ] Cos[\[Psi]]Cos[ψ]}};

mt2 = {{a, b, c}, {b, d, e}, {c, e, h}};

Here mt1mt1 is an orthogonal matrix and mt2mt2 is a real symmetric matrix. In principle, the output of Transpose[mt1].mt2.mt1 should be a diagonal matrix, but Mathematica cannot do it. Any suggestion is appreciated!

mt1 = {{Cos[\[Xi]] Cos[\[Phi]] - Sin[\[Xi]] Sin[\[Phi]] Sin[\[Psi]], 
    Cos[\[Psi]] Sin[\[Phi]], 
    Cos[\[Phi]] Sin[\[Xi]] + 
     Cos[\[Xi]] Sin[\[Phi]] Sin[\[Psi]]}, {-Cos[\[Xi]] Sin[\[Phi]] - 
     Cos[\[Phi]] Sin[\[Xi]] Sin[\[Psi]], 
    Cos[\[Phi]] Cos[\[Psi]], -Sin[\[Xi]] Sin[\[Phi]] + 
      Cos[\[Xi]] Cos[\[Phi]] Sin[\[Psi]]}, {-Cos[\[Psi]] Sin[\[Xi]], \
-Sin[\[Psi]], Cos[\[Xi]] Cos[\[Psi]]}};

mt2 = {{a, b, c}, {b, d, e}, {c, e, h}};

Here mt1 is an orthogonal matrix and mt2 is a real symmetric matrix. In principle, the output of Transpose[mt1].mt2.mt1 should be a diagonal matrix, but Mathematica cannot do it. Any suggestion is appreciated!

mt1 = {{Cos[ξ] Cos[ϕ] - Sin[ξ] Sin[ϕ] Sin[ψ], Cos[ψ] Sin[ϕ], Cos[ϕ] Sin[ξ] + Cos[ξ] Sin[ϕ] Sin[ψ]},
       {-Cos[ξ] Sin[ϕ] - Cos[ϕ] Sin[ξ] Sin[ψ], Cos[ϕ] Cos[ψ], -Sin[ξ] Sin[ϕ] + Cos[ξ] Cos[ϕ] Sin[ψ]},
       {-Cos[ψ] Sin[ξ], -Sin[ψ], Cos[ξ] Cos[ψ]}};

mt2 = {{a, b, c}, {b, d, e}, {c, e, h}};

Here mt1 is an orthogonal matrix and mt2 is a real symmetric matrix. In principle, the output of Transpose[mt1].mt2.mt1 should be a diagonal matrix, but Mathematica cannot do it. Any suggestion is appreciated!

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SciJewel
SciJewel
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Diagonalizing a real symmetric matrix with an orthogonal matrix

mt1 = {{Cos[\[Xi]] Cos[\[Phi]] - Sin[\[Xi]] Sin[\[Phi]] Sin[\[Psi]], 
    Cos[\[Psi]] Sin[\[Phi]], 
    Cos[\[Phi]] Sin[\[Xi]] + 
     Cos[\[Xi]] Sin[\[Phi]] Sin[\[Psi]]}, {-Cos[\[Xi]] Sin[\[Phi]] - 
     Cos[\[Phi]] Sin[\[Xi]] Sin[\[Psi]], 
    Cos[\[Phi]] Cos[\[Psi]], -Sin[\[Xi]] Sin[\[Phi]] + 
     Cos[\[Xi]] Cos[\[Phi]] Sin[\[Psi]]}, {-Cos[\[Psi]] Sin[\[Xi]], \
-Sin[\[Psi]], Cos[\[Xi]] Cos[\[Psi]]}};

mt2 = {{a, b, c}, {b, d, e}, {c, e, h}};

Here mt1 is an orthogonal matrix and mt2 is a real symmetric matrix. In principle, the output of Transpose[mt1].mt2.mt1 should be a diagonal matrix, but Mathematica cannot do it. Any suggestion is appreciated!