Orthogonal Diagonalization
Selected Homework 10 Answers, Exercise 1.
![A = matrix([[-2, 0, -36], [0, -3, 0], [-36, 0, -23]]), ` `*(A-I*lambda) = matrix([[-2-lambda, 0, -36], [0, -3-lambda, 0], [-36, 0, -23-lambda]])](images/OrthogonalDiagSHW10Ex1Ans2.gif)
![evecsAvec1 = vector([-4/3, 0, 1])](images/OrthogonalDiagSHW10Ex1Ans6.gif){kind=small}
![A-I*lambda = matrix([[-27, 0, -36], [0, -28, 0], [-36, 0, -48]])](images/OrthogonalDiagSHW10Ex1Ans8.gif)
![matrix([[-27, 0, -36], [0, -28, 0], [-36, 0, -48]])*matrix([[a], [b], [c]]) = matrix([[0], [0], [0]])](images/OrthogonalDiagSHW10Ex1Ans9.gif)
![matrix([[1, 0, 4/3], [0, 1, 0], [0, 0, 0]])*matrix([[a], [b], [c]]) = matrix([[0], [0], [0]])](images/OrthogonalDiagSHW10Ex1Ans10.gif)
![matrix([[a+4/3*c], [b], [0]]) = matrix([[0], [0], [0]])](images/OrthogonalDiagSHW10Ex1Ans11.gif)
![v[1] = matrix([[-4/3], [0], [1]])](images/OrthogonalDiagSHW10Ex1Ans12.gif)
![evecsAvec2 = vector([0, 1, 0])](images/OrthogonalDiagSHW10Ex1Ans15.gif){kind=small}
![A-I*lambda = matrix([[1, 0, -36], [0, 0, 0], [-36, 0, -20]])](images/OrthogonalDiagSHW10Ex1Ans17.gif)
![matrix([[1, 0, -36], [0, 0, 0], [-36, 0, -20]])*matrix([[a], [b], [c]]) = matrix([[0], [0], [0]])](images/OrthogonalDiagSHW10Ex1Ans18.gif)
![matrix([[1, 0, 0], [0, 0, 1], [0, 0, 0]])*matrix([[a], [b], [c]]) = matrix([[0], [0], [0]])](images/OrthogonalDiagSHW10Ex1Ans19.gif)
![matrix([[a], [c], [0]]) = matrix([[0], [0], [0]])](images/OrthogonalDiagSHW10Ex1Ans20.gif)
![v[2] = matrix([[0], [1], [0]])](images/OrthogonalDiagSHW10Ex1Ans21.gif)
![evecsAvec3 = vector([1, 0, 4/3])](images/OrthogonalDiagSHW10Ex1Ans24.gif){kind=small}
![A-I*lambda = matrix([[48, 0, -36], [0, 47, 0], [-36, 0, 27]])](images/OrthogonalDiagSHW10Ex1Ans26.gif)
![matrix([[48, 0, -36], [0, 47, 0], [-36, 0, 27]])*matrix([[a], [b], [c]]) = matrix([[0], [0], [0]])](images/OrthogonalDiagSHW10Ex1Ans27.gif)
![matrix([[1, 0, -3/4], [0, 1, 0], [0, 0, 0]])*matrix([[a], [b], [c]]) = matrix([[0], [0], [0]])](images/OrthogonalDiagSHW10Ex1Ans28.gif)
![matrix([[a-3/4*c], [b], [0]]) = matrix([[0], [0], [0]])](images/OrthogonalDiagSHW10Ex1Ans29.gif)
![v[3] = matrix([[1], [0], [4/3]])](images/OrthogonalDiagSHW10Ex1Ans30.gif)
![v[1] = matrix([[-4/3], [0], [1]]), v[2] = matrix([[0], [1], [0]]), v[3] = matrix([[1], [0], [4/3]])](images/OrthogonalDiagSHW10Ex1Ans32.gif)
![P = matrix([[-4/3, 0, 1], [0, 1, 0], [1, 0, 4/3]]), P^`-1` = matrix([[-12/25, 0, 9/25], [0, 1, 0], [9/25, 0, 12/25]])](images/OrthogonalDiagSHW10Ex1Ans33.gif)
![P^`-1`*P*` = `*matrix([[-12/25, 0, 9/25], [0, 1, 0], [9/25, 0, 12/25]])*matrix([[-4/3, 0, 1], [0, 1, 0], [1, 0, 4/3]]) = matrix([[1, 0, 0], [0, 1, 0], [0, 0, 1]])](images/OrthogonalDiagSHW10Ex1Ans34.gif)
![P^`-1`*A*P*` = `*matrix([[-12/25, 0, 9/25], [0, 1, 0], [9/25, 0, 12/25]])*matrix([[-2, 0, -36], [0, -3, 0], [-36, 0, -23]])*matrix([[-4/3, 0, 1], [0, 1, 0], [1, 0, 4/3]]) = matrix([[25, 0, 0], [0, -3, ...](images/OrthogonalDiagSHW10Ex1Ans35.gif)
![v[1] = vector([-4/3, 0, 1])](images/OrthogonalDiagSHW10Ex1Ans38.gif){kind=small}
![v[2] = vector([0, 1, 0])](images/OrthogonalDiagSHW10Ex1Ans39.gif){kind=small}
![v[3] = vector([1, 0, 4/3])](images/OrthogonalDiagSHW10Ex1Ans40.gif){kind=small}
![w[1] = v[1]](images/OrthogonalDiagSHW10Ex1Ans42.gif){kind=small}
![w[2] = v[2]-`(`*v[2]*` dot`*` w`[1]*`)`/` ||`/w[1]/`||`^2*`w `[1]](images/OrthogonalDiagSHW10Ex1Ans43.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/OrthogonalDiagSHW10Ex1Ans44.gif)
![w[1] = vector([-4/3, 0, 1])](images/OrthogonalDiagSHW10Ex1Ans45.gif){kind=small}
![w[2] = vector([0, 1, 0])-`(`*vector([0, 1, 0])*` dot`*vector([-4/3, 0, 1])*`)`/` ||`/vector([-4/3, 0, 1])/`||`^2*vector([-4/3, 0, 1])](images/OrthogonalDiagSHW10Ex1Ans46.gif)
![w[3] = vector([1, 0, 4/3])-`(`*vector([1, 0, 4/3])*` dot`*vector([-4/3, 0, 1])*`)`/` ||`/vector([-4/3, 0, 1])/`||`^2*vector([-4/3, 0, 1])-`(`*vector([1, 0, 4/3])*` dot`*vector([0, 1, 0])*`)`/` ||`/vect...](images/OrthogonalDiagSHW10Ex1Ans47.gif)
![w[1] = vector([-4/3, 0, 1])](images/OrthogonalDiagSHW10Ex1Ans49.gif){kind=small}
![w[2] = vector([0, 1, 0])](images/OrthogonalDiagSHW10Ex1Ans50.gif){kind=small}
![w[3] = vector([1, 0, 4/3])](images/OrthogonalDiagSHW10Ex1Ans51.gif){kind=small}
![u[1] = 1/` ||`/w[1]/`||`*` w`[1], ` ||`*w[1]*`||` = 5/3](images/OrthogonalDiagSHW10Ex1Ans53.gif)
![u[2] = 1/` ||`/w[2]/`||`*` w`[2], ` ||`*w[2]*`||` = 1](images/OrthogonalDiagSHW10Ex1Ans54.gif)
![u[3] = 1/` ||`/w[3]/`||`*` w`[3], ` ||`*w[3]*`||` = 5/3](images/OrthogonalDiagSHW10Ex1Ans55.gif)
![u[1] = matrix([[-4/5], [0], [3/5]]), u[2] = matrix([[0], [1], [0]]), u[3] = matrix([[3/5], [0], [4/5]])](images/OrthogonalDiagSHW10Ex1Ans56.gif)
![Q = matrix([[-4/5, 0, 3/5], [0, 1, 0], [3/5, 0, 4/5]]), Q^`-1` = matrix([[-4/5, 0, 3/5], [0, 1, 0], [3/5, 0, 4/5]])](images/OrthogonalDiagSHW10Ex1Ans58.gif)
![Q^`-1`*Q*` = `*matrix([[-4/5, 0, 3/5], [0, 1, 0], [3/5, 0, 4/5]])*matrix([[-4/5, 0, 3/5], [0, 1, 0], [3/5, 0, 4/5]]) = matrix([[1, 0, 0], [0, 1, 0], [0, 0, 1]])](images/OrthogonalDiagSHW10Ex1Ans59.gif)
![Q^`-1`*A*Q*` = `*matrix([[-4/5, 0, 3/5], [0, 1, 0], [3/5, 0, 4/5]])*matrix([[-2, 0, -36], [0, -3, 0], [-36, 0, -23]])*matrix([[-4/5, 0, 3/5], [0, 1, 0], [3/5, 0, 4/5]]) = matrix([[25, 0, 0], [0, -3, 0]...](images/OrthogonalDiagSHW10Ex1Ans60.gif)
