Orthogonal
Diagonalization
Example 4.
(
Eigenvalues
& Eigenvectors Example 6.)
![v[1] = matrix([[-1], [0], [1]]), v[2] = matrix([[-1], [1], [0]]), v[3] = matrix([[1], [1], [1]])](images/OrthogonalDiagExample42.gif)
![P = matrix([[-1, -1, 1], [0, 1, 1], [1, 0, 1]]), P^`-1` = matrix([[-1/3, -1/3, 2/3], [-1/3, 2/3, -1/3], [1/3, 1/3, 1/3]])](images/OrthogonalDiagExample43.gif)
![P^`-1`*P*` = `*matrix([[-1/3, -1/3, 2/3], [-1/3, 2/3, -1/3], [1/3, 1/3, 1/3]])*matrix([[-1, -1, 1], [0, 1, 1], [1, 0, 1]]) = matrix([[1, 0, 0], [0, 1, 0], [0, 0, 1]])](images/OrthogonalDiagExample44.gif)
![P^`-1`*A*P*` = `*matrix([[-1/3, -1/3, 2/3], [-1/3, 2/3, -1/3], [1/3, 1/3, 1/3]])*matrix([[4, 2, 2], [2, 4, 2], [2, 2, 4]])*matrix([[-1, -1, 1], [0, 1, 1], [1, 0, 1]]) = matrix([[2, 0, 0], [0, 2, 0], [0...](images/OrthogonalDiagExample45.gif)
![v[1] = vector([-1, 0, 1])](images/OrthogonalDiagExample49.gif){kind=small}
![v[2] = vector([-1, 1, 0])](images/OrthogonalDiagExample410.gif){kind=small}
![v[3] = vector([1, 1, 1])](images/OrthogonalDiagExample411.gif){kind=small}
![w[1] = v[1]](images/OrthogonalDiagExample413.gif){kind=small}
![w[2] = v[2]-`(`*v[2]*` dot`*` w`[1]*`)`/` ||`/w[1]/`||`^2*`w `[1]](images/OrthogonalDiagExample414.gif)
![w[3] = v[3]-`(`*v[3]*` dot`*` w`[1]*`)`/` ||`/w[1]/`||`^2*`w `[1]-`(`*v[3]*` dot`*` w`[2]*`)`/` ||`/w[2]/`||`^2*`w `[2]](images/OrthogonalDiagExample415.gif)
![w[1] = vector([-1, 0, 1])](images/OrthogonalDiagExample416.gif){kind=small}
![w[2] = vector([-1, 1, 0])-`(`*vector([-1, 1, 0])*` dot`*vector([-1, 0, 1])*`)`/` ||`/vector([-1, 0, 1])/`||`^2*vector([-1, 0, 1])](images/OrthogonalDiagExample417.gif)
![w[3] = vector([1, 1, 1])-`(`*vector([1, 1, 1])*` dot`*vector([-1, 0, 1])*`)`/` ||`/vector([-1, 0, 1])/`||`^2*vector([-1, 0, 1])-`(`*vector([1, 1, 1])*` dot`*vector([-1/2, 1, -1/2])*`)`/` ||`/vector([-1...](images/OrthogonalDiagExample418.gif)
![w[1] = vector([-1, 0, 1])](images/OrthogonalDiagExample420.gif){kind=small}
![w[2] = vector([-1/2, 1, -1/2])](images/OrthogonalDiagExample421.gif){kind=small}
![w[3] = vector([1, 1, 1])](images/OrthogonalDiagExample422.gif){kind=small}
![u[1] = 1/` ||`/w[1]/`||`*` w`[1], ` ||`*w[1]*`||` = 2^(1/2)](images/OrthogonalDiagExample424.gif)
![u[2] = 1/` ||`/w[2]/`||`*` w`[2], ` ||`*w[2]*`||` = 1/2*6^(1/2)](images/OrthogonalDiagExample425.gif)
![u[3] = 1/` ||`/w[3]/`||`*` w`[3], ` ||`*w[3]*`||` = 3^(1/2)](images/OrthogonalDiagExample426.gif)
![u[1] = matrix([[-1/2*2^(1/2)], [0], [1/2*2^(1/2)]]), u[2] = matrix([[-1/6*6^(1/2)], [1/3*6^(1/2)], [-1/6*6^(1/2)]]), u[3] = matrix([[1/3*3^(1/2)], [1/3*3^(1/2)], [1/3*3^(1/2)]])](images/OrthogonalDiagExample427.gif)
![Q = matrix([[-1/2*2^(1/2), -1/6*6^(1/2), 1/3*3^(1/2)], [0, 1/3*6^(1/2), 1/3*3^(1/2)], [1/2*2^(1/2), -1/6*6^(1/2), 1/3*3^(1/2)]]), Q^`-1` = matrix([[-1/2*2^(1/2), 0, 1/2*2^(1/2)], [-1/6*6^(1/2), 1/3*6^(...](images/OrthogonalDiagExample429.gif)
![Q^`-1`*Q*` = `*matrix([[-1/2*2^(1/2), 0, 1/2*2^(1/2)], [-1/6*6^(1/2), 1/3*6^(1/2), -1/6*6^(1/2)], [1/3*3^(1/2), 1/3*3^(1/2), 1/3*3^(1/2)]])*matrix([[-1/2*2^(1/2), -1/6*6^(1/2), 1/3*3^(1/2)], [0, 1/3*6^...](images/OrthogonalDiagExample430.gif)
![Q^`-1`*A*Q*` = `*matrix([[-1/2*2^(1/2), 0, 1/2*2^(1/2)], [-1/6*6^(1/2), 1/3*6^(1/2), -1/6*6^(1/2)], [1/3*3^(1/2), 1/3*3^(1/2), 1/3*3^(1/2)]])*matrix([[4, 2, 2], [2, 4, 2], [2, 2, 4]])*matrix([[-1/2*2^(...](images/OrthogonalDiagExample431.gif)
