Orthogonal Diagonalization
Selected Homework 10 Answers, Exercise 4.
![A = matrix([[3, 1, 0, 0], [1, 3, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0]]), ` `*(A-I*lambda) = matrix([[3-lambda, 1, 0, 0], [1, 3-lambda, 0, 0], [0, 0, -lambda, 0], [0, 0, 0, -lambda]])](images/OrthogonalDiagSHW10Ex4Ans169.gif)
![evecsAvec1 = vector([0, 0, 0, 1])](images/OrthogonalDiagSHW10Ex4Ans173.gif){kind=small}
![evecsAvec12 = vector([0, 0, 1, 0])](images/OrthogonalDiagSHW10Ex4Ans174.gif){kind=small}
![A-I*lambda = matrix([[3, 1, 0, 0], [1, 3, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0]])](images/OrthogonalDiagSHW10Ex4Ans176.gif)
![matrix([[3, 1, 0, 0], [1, 3, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0]])*matrix([[a], [b], [c], [d]]) = matrix([[0], [0], [0], [0]])](images/OrthogonalDiagSHW10Ex4Ans177.gif)
![matrix([[1, 0, 0, 0], [0, 1, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0]])*matrix([[a], [b], [c], [d]]) = matrix([[0], [0], [0], [0]])](images/OrthogonalDiagSHW10Ex4Ans178.gif)
![matrix([[a], [b], [0], [0]]) = matrix([[0], [0], [0], [0]])](images/OrthogonalDiagSHW10Ex4Ans179.gif)
![v[1] = matrix([[0], [0], [0], [1]]), v[2] = matrix([[0], [0], [1], [0]])](images/OrthogonalDiagSHW10Ex4Ans180.gif)
![evecsAvec2 = vector([-1, 1, 0, 0])](images/OrthogonalDiagSHW10Ex4Ans183.gif){kind=small}
![A-I*lambda = matrix([[1, 1, 0, 0], [1, 1, 0, 0], [0, 0, -2, 0], [0, 0, 0, -2]])](images/OrthogonalDiagSHW10Ex4Ans185.gif)
![matrix([[1, 1, 0, 0], [1, 1, 0, 0], [0, 0, -2, 0], [0, 0, 0, -2]])*matrix([[a], [b], [c], [d]]) = matrix([[0], [0], [0], [0]])](images/OrthogonalDiagSHW10Ex4Ans186.gif)
![matrix([[1, 1, 0, 0], [0, 0, 1, 0], [0, 0, 0, 1], [0, 0, 0, 0]])*matrix([[a], [b], [c], [d]]) = matrix([[0], [0], [0], [0]])](images/OrthogonalDiagSHW10Ex4Ans187.gif)
![matrix([[a+b], [c], [d], [0]]) = matrix([[0], [0], [0], [0]])](images/OrthogonalDiagSHW10Ex4Ans188.gif)
![v[3] = matrix([[-1], [1], [0], [0]])](images/OrthogonalDiagSHW10Ex4Ans189.gif)
![evecsAvec3 = vector([1, 1, 0, 0])](images/OrthogonalDiagSHW10Ex4Ans192.gif){kind=small}
![A-I*lambda = matrix([[-1, 1, 0, 0], [1, -1, 0, 0], [0, 0, -4, 0], [0, 0, 0, -4]])](images/OrthogonalDiagSHW10Ex4Ans194.gif)
![matrix([[-1, 1, 0, 0], [1, -1, 0, 0], [0, 0, -4, 0], [0, 0, 0, -4]])*matrix([[a], [b], [c], [d]]) = matrix([[0], [0], [0], [0]])](images/OrthogonalDiagSHW10Ex4Ans195.gif)
![matrix([[1, -1, 0, 0], [0, 0, 1, 0], [0, 0, 0, 1], [0, 0, 0, 0]])*matrix([[a], [b], [c], [d]]) = matrix([[0], [0], [0], [0]])](images/OrthogonalDiagSHW10Ex4Ans196.gif)
![matrix([[a-b], [c], [d], [0]]) = matrix([[0], [0], [0], [0]])](images/OrthogonalDiagSHW10Ex4Ans197.gif)
![v[4] = matrix([[1], [1], [0], [0]])](images/OrthogonalDiagSHW10Ex4Ans198.gif)
![v[1] = matrix([[0], [0], [0], [1]]), v[2] = matrix([[0], [0], [1], [0]]), v[3] = matrix([[-1], [1], [0], [0]]), v[4] = matrix([[1], [1], [0], [0]])](images/OrthogonalDiagSHW10Ex4Ans200.gif)
![P = matrix([[0, 0, -1, 1], [0, 0, 1, 1], [0, 1, 0, 0], [1, 0, 0, 0]]), P^`-1` = matrix([[0, 0, 0, 1], [0, 0, 1, 0], [-1/2, 1/2, 0, 0], [1/2, 1/2, 0, 0]])](images/OrthogonalDiagSHW10Ex4Ans201.gif)
![P^`-1`*P*` = `*matrix([[0, 0, 0, 1], [0, 0, 1, 0], [-1/2, 1/2, 0, 0], [1/2, 1/2, 0, 0]])*matrix([[0, 0, -1, 1], [0, 0, 1, 1], [0, 1, 0, 0], [1, 0, 0, 0]]) = matrix([[1, 0, 0, 0], [0, 1, 0, 0], [0, 0, 1...](images/OrthogonalDiagSHW10Ex4Ans202.gif)
![P^`-1`*A*P*` = `*matrix([[0, 0, 0, 1], [0, 0, 1, 0], [-1/2, 1/2, 0, 0], [1/2, 1/2, 0, 0]])*matrix([[3, 1, 0, 0], [1, 3, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0]])*matrix([[0, 0, -1, 1], [0, 0, 1, 1], [0, 1, 0...](images/OrthogonalDiagSHW10Ex4Ans203.gif)
![v[1] = matrix([[0], [0], [0], [1]]), v[2] = matrix([[0], [0], [1], [0]]), v[3] = matrix([[-1], [1], [0], [0]]), v[4] = matrix([[1], [1], [0], [0]])](images/OrthogonalDiagSHW10Ex4Ans205.gif)
![P = matrix([[0, 0, -1, 1], [0, 0, 1, 1], [0, 1, 0, 0], [1, 0, 0, 0]]), P^`-1` = matrix([[0, 0, 0, 1], [0, 0, 1, 0], [-1/2, 1/2, 0, 0], [1/2, 1/2, 0, 0]])](images/OrthogonalDiagSHW10Ex4Ans206.gif)
![P^`-1`*P*` = `*matrix([[0, 0, 0, 1], [0, 0, 1, 0], [-1/2, 1/2, 0, 0], [1/2, 1/2, 0, 0]])*matrix([[0, 0, -1, 1], [0, 0, 1, 1], [0, 1, 0, 0], [1, 0, 0, 0]]) = matrix([[1, 0, 0, 0], [0, 1, 0, 0], [0, 0, 1...](images/OrthogonalDiagSHW10Ex4Ans207.gif)
![P^`-1`*A*P*` = `*matrix([[0, 0, 0, 1], [0, 0, 1, 0], [-1/2, 1/2, 0, 0], [1/2, 1/2, 0, 0]])*matrix([[3, 1, 0, 0], [1, 3, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0]])*matrix([[0, 0, -1, 1], [0, 0, 1, 1], [0, 1, 0...](images/OrthogonalDiagSHW10Ex4Ans208.gif)
![v[1] = vector([0, 0, 0, 1])](images/OrthogonalDiagSHW10Ex4Ans212.gif){kind=small}
![v[2] = vector([0, 0, 1, 0])](images/OrthogonalDiagSHW10Ex4Ans213.gif){kind=small}
![v[3] = vector([-1, 1, 0, 0])](images/OrthogonalDiagSHW10Ex4Ans214.gif){kind=small}
![v[4] = vector([1, 1, 0, 0])](images/OrthogonalDiagSHW10Ex4Ans215.gif){kind=small}
![w[1] = v[1]](images/OrthogonalDiagSHW10Ex4Ans217.gif){kind=small}
![w[2] = v[2]-`(`*v[2]*` dot`*` w`[1]*`)`/` ||`/w[1]/`||`^2*`w `[1]](images/OrthogonalDiagSHW10Ex4Ans218.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/OrthogonalDiagSHW10Ex4Ans219.gif)
![w[4] = v[4]-`(`*v[4]*` dot`*` w`[1]*`)`/` ||`/w[1]/`||`^2*`w `[1]-`(`*v[4]*` dot`*` w`[2]*`)`/` ||`/w[2]/`||`^2*`w `[2]-`(`*v[4]*` dot`*` w`[3]*`)`/` ||`/w[3]/`||`^2*`w `[3]](images/OrthogonalDiagSHW10Ex4Ans220.gif)
![w[1] = vector([0, 0, 0, 1])](images/OrthogonalDiagSHW10Ex4Ans221.gif){kind=small}
![w[2] = vector([0, 0, 1, 0])-`(`*vector([0, 0, 1, 0])*` dot`*vector([0, 0, 0, 1])*`)`/` ||`/vector([0, 0, 0, 1])/`||`^2*vector([0, 0, 0, 1])](images/OrthogonalDiagSHW10Ex4Ans222.gif)
![w[3] = vector([-1, 1, 0, 0])-`(`*vector([-1, 1, 0, 0])*` dot`*vector([0, 0, 0, 1])*`)`/` ||`/vector([0, 0, 0, 1])/`||`^2*vector([0, 0, 0, 1])-`(`*vector([-1, 1, 0, 0])*` dot`*vector([0, 0, 1, 0])*`)`/`...](images/OrthogonalDiagSHW10Ex4Ans223.gif)
![w[4] = (vector([1, 1, 0, 0])-`(`*vector([1, 1, 0, 0])*` dot`*vector([0, 0, 0, 1])*`)`/` ||`/vector([0, 0, 0, 1])/`||`^2*vector([0, 0, 0, 1])-`(`*vector([1, 1, 0, 0])*` dot`*vector([0, 0, 1, 0])*`)`/` |...](images/OrthogonalDiagSHW10Ex4Ans224.gif){kind=small}
![w[4] = (vector([1, 1, 0, 0])-`(`*vector([1, 1, 0, 0])*` dot`*vector([0, 0, 0, 1])*`)`/` ||`/vector([0, 0, 0, 1])/`||`^2*vector([0, 0, 0, 1])-`(`*vector([1, 1, 0, 0])*` dot`*vector([0, 0, 1, 0])*`)`/` |...](images/OrthogonalDiagSHW10Ex4Ans225.gif)
![w[1] = vector([0, 0, 0, 1])](images/OrthogonalDiagSHW10Ex4Ans227.gif){kind=small}
![w[2] = vector([0, 0, 1, 0])](images/OrthogonalDiagSHW10Ex4Ans228.gif){kind=small}
![w[3] = vector([-1, 1, 0, 0])](images/OrthogonalDiagSHW10Ex4Ans229.gif){kind=small}
![w[4] = vector([1, 1, 0, 0])](images/OrthogonalDiagSHW10Ex4Ans230.gif){kind=small}
![u[1] = 1/` ||`/w[1]/`||`*` w`[1], ` ||`*w[1]*`||` = 1](images/OrthogonalDiagSHW10Ex4Ans232.gif)
![u[2] = 1/` ||`/w[2]/`||`*` w`[2], ` ||`*w[2]*`||` = 1](images/OrthogonalDiagSHW10Ex4Ans233.gif)
![u[3] = 1/` ||`/w[3]/`||`*` w`[3], ` ||`*w[3]*`||` = 2^(1/2)](images/OrthogonalDiagSHW10Ex4Ans234.gif)
![u[4] = 1/` ||`/w[4]/`||`*` w`[4], ` ||`*w[4]*`||` = 2^(1/2)](images/OrthogonalDiagSHW10Ex4Ans235.gif)
![u[1] = matrix([[0], [0], [0], [1]]), u[2] = matrix([[0], [0], [1], [0]]), u[3] = matrix([[-1/2*2^(1/2)], [1/2*2^(1/2)], [0], [0]]), u[4] = matrix([[1/2*2^(1/2)], [1/2*2^(1/2)], [0], [0]])](images/OrthogonalDiagSHW10Ex4Ans236.gif)
![Q = matrix([[0, 0, -1/2*2^(1/2), 1/2*2^(1/2)], [0, 0, 1/2*2^(1/2), 1/2*2^(1/2)], [0, 1, 0, 0], [1, 0, 0, 0]]), Q^`-1` = matrix([[0, 0, 0, 1], [0, 0, 1, 0], [-1/2*2^(1/2), 1/2*2^(1/2), 0, 0], [1/2*2^(1/...](images/OrthogonalDiagSHW10Ex4Ans238.gif)
![Q^`-1`*Q*` = `*matrix([[0, 0, 0, 1], [0, 0, 1, 0], [-1/2*2^(1/2), 1/2*2^(1/2), 0, 0], [1/2*2^(1/2), 1/2*2^(1/2), 0, 0]])*matrix([[0, 0, -1/2*2^(1/2), 1/2*2^(1/2)], [0, 0, 1/2*2^(1/2), 1/2*2^(1/2)], [0,...](images/OrthogonalDiagSHW10Ex4Ans239.gif)
![Q^`-1`*A*Q*` = `*matrix([[0, 0, 0, 1], [0, 0, 1, 0], [-1/2*2^(1/2), 1/2*2^(1/2), 0, 0], [1/2*2^(1/2), 1/2*2^(1/2), 0, 0]])*matrix([[3, 1, 0, 0], [1, 3, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0]])*matrix([[0, 0...](images/OrthogonalDiagSHW10Ex4Ans240.gif)
