Orthogonal
Diagonalization
Example 3.
(
Eigenvalues
& Eigenvectors Example 5.)
![v[1] = matrix([[0], [1], [0]]), v[2] = matrix([[-1], [0], [1]]), v[3] = matrix([[-2], [1], [1]])](images/OrthogonalDiagExample32.gif)
![P = matrix([[0, -1, -2], [1, 0, 1], [0, 1, 1]]), P^`-1` = matrix([[1, 1, 1], [1, 0, 2], [-1, 0, -1]])](images/OrthogonalDiagExample33.gif)
![P^`-1`*P*` = `*matrix([[1, 1, 1], [1, 0, 2], [-1, 0, -1]])*matrix([[0, -1, -2], [1, 0, 1], [0, 1, 1]]) = matrix([[1, 0, 0], [0, 1, 0], [0, 0, 1]])](images/OrthogonalDiagExample34.gif)
![P^`-1`*A*P*` = `*matrix([[1, 1, 1], [1, 0, 2], [-1, 0, -1]])*matrix([[0, 0, -2], [1, 2, 1], [1, 0, 3]])*matrix([[0, -1, -2], [1, 0, 1], [0, 1, 1]]) = matrix([[2, 0, 0], [0, 2, 0], [0, 0, 1]])](images/OrthogonalDiagExample35.gif)
![v[1] = vector([0, 1, 0])](images/OrthogonalDiagExample39.gif){kind=small}
![v[2] = vector([-1, 0, 1])](images/OrthogonalDiagExample310.gif){kind=small}
![v[3] = vector([-2, 1, 1])](images/OrthogonalDiagExample311.gif){kind=small}
![w[1] = v[1]](images/OrthogonalDiagExample313.gif){kind=small}
![w[2] = v[2]-`(`*v[2]*` dot`*` w`[1]*`)`/` ||`/w[1]/`||`^2*`w `[1]](images/OrthogonalDiagExample314.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/OrthogonalDiagExample315.gif)
![w[1] = vector([0, 1, 0])](images/OrthogonalDiagExample316.gif){kind=small}
![w[2] = vector([-1, 0, 1])-`(`*vector([-1, 0, 1])*` dot`*vector([0, 1, 0])*`)`/` ||`/vector([0, 1, 0])/`||`^2*vector([0, 1, 0])](images/OrthogonalDiagExample317.gif)
![w[3] = vector([-2, 1, 1])-`(`*vector([-2, 1, 1])*` dot`*vector([0, 1, 0])*`)`/` ||`/vector([0, 1, 0])/`||`^2*vector([0, 1, 0])-`(`*vector([-2, 1, 1])*` dot`*vector([-1, 0, 1])*`)`/` ||`/vector([-1, 0, ...](images/OrthogonalDiagExample318.gif)
![w[1] = vector([0, 1, 0])](images/OrthogonalDiagExample320.gif){kind=small}
![w[2] = vector([-1, 0, 1])](images/OrthogonalDiagExample321.gif){kind=small}
![w[3] = vector([-1/2, 0, -1/2])](images/OrthogonalDiagExample322.gif){kind=small}
![u[1] = 1/` ||`/w[1]/`||`*` w`[1], ` ||`*w[1]*`||` = 1](images/OrthogonalDiagExample324.gif)
![u[2] = 1/` ||`/w[2]/`||`*` w`[2], ` ||`*w[2]*`||` = 2^(1/2)](images/OrthogonalDiagExample325.gif)
![u[3] = 1/` ||`/w[3]/`||`*` w`[3], ` ||`*w[3]*`||` = 1/2*2^(1/2)](images/OrthogonalDiagExample326.gif)
![u[1] = matrix([[0], [1], [0]]), u[2] = matrix([[-1/2*2^(1/2)], [0], [1/2*2^(1/2)]]), u[3] = matrix([[-1/2*2^(1/2)], [0], [-1/2*2^(1/2)]])](images/OrthogonalDiagExample327.gif)
![Q = matrix([[0, -1/2*2^(1/2), -1/2*2^(1/2)], [1, 0, 0], [0, 1/2*2^(1/2), -1/2*2^(1/2)]]), Q^`-1` = matrix([[0, 1, 0], [-1/2*2^(1/2), 0, 1/2*2^(1/2)], [-1/2*2^(1/2), 0, -1/2*2^(1/2)]])](images/OrthogonalDiagExample329.gif)
![Q^`-1`*Q*` = `*matrix([[0, 1, 0], [-1/2*2^(1/2), 0, 1/2*2^(1/2)], [-1/2*2^(1/2), 0, -1/2*2^(1/2)]])*matrix([[0, -1/2*2^(1/2), -1/2*2^(1/2)], [1, 0, 0], [0, 1/2*2^(1/2), -1/2*2^(1/2)]]) = matrix([[1, 0,...](images/OrthogonalDiagExample330.gif)
![Q^`-1`*A*Q*` = `*matrix([[0, 1, 0], [-1/2*2^(1/2), 0, 1/2*2^(1/2)], [-1/2*2^(1/2), 0, -1/2*2^(1/2)]])*matrix([[0, 0, -2], [1, 2, 1], [1, 0, 3]])*matrix([[0, -1/2*2^(1/2), -1/2*2^(1/2)], [1, 0, 0], [0, ...](images/OrthogonalDiagExample331.gif)
