Orthogonal Diagonalization

 

Selected Homework 10 Answers,  Exercise 5.

`Exercise 5.  Eigenvalue, eigenvector analysis of `*A*`:`

A = matrix([[-7, 24, 0, 0], [24, 7, 0, 0], [0, 0, -7, 24], [0, 0, 24, 7]]), `  `*(A-I*lambda) = matrix([[-7-lambda, 24, 0, 0], [24, 7-lambda, 0, 0], [0, 0, -7-lambda, 24], [0, 0, 24, 7-lambda]])

det(A-I*lambda)*` = `*(lambda-25)^2*(lambda+25)^2 = 0

evecsAval1 = 25

evecsAmul1 = 2

evecsAvec1 = vector([1, 4/3, 0, 0])

evecsAvec12 = vector([0, 0, 1, 4/3])

lambda = 25

A-I*lambda = matrix([[-32, 24, 0, 0], [24, -18, 0, 0], [0, 0, -32, 24], [0, 0, 24, -18]])

matrix([[-32, 24, 0, 0], [24, -18, 0, 0], [0, 0, -32, 24], [0, 0, 24, -18]])*matrix([[a], [b], [c], [d]]) = matrix([[0], [0], [0], [0]])

matrix([[1, -3/4, 0, 0], [0, 0, 1, -3/4], [0, 0, 0, 0], [0, 0, 0, 0]])*matrix([[a], [b], [c], [d]]) = matrix([[0], [0], [0], [0]])

matrix([[a-3/4*b], [c-3/4*d], [0], [0]]) = matrix([[0], [0], [0], [0]])

v[1] = matrix([[1], [4/3], [0], [0]]), v[2] = matrix([[0], [0], [1], [4/3]])

evecsAval2 = -25

evecsAmul2 = 2

evecsAvec2 = vector([-4/3, 1, 0, 0])

evecsAvec22 = vector([0, 0, -4/3, 1])

lambda = -25

A-I*lambda = matrix([[18, 24, 0, 0], [24, 32, 0, 0], [0, 0, 18, 24], [0, 0, 24, 32]])

matrix([[18, 24, 0, 0], [24, 32, 0, 0], [0, 0, 18, 24], [0, 0, 24, 32]])*matrix([[a], [b], [c], [d]]) = matrix([[0], [0], [0], [0]])

matrix([[1, 4/3, 0, 0], [0, 0, 1, 4/3], [0, 0, 0, 0], [0, 0, 0, 0]])*matrix([[a], [b], [c], [d]]) = matrix([[0], [0], [0], [0]])

matrix([[a+4/3*b], [c+4/3*d], [0], [0]]) = matrix([[0], [0], [0], [0]])

v[3] = matrix([[-4/3], [1], [0], [0]]), v[4] = matrix([[0], [0], [-4/3], [1]])

`A is diagonalizable using the eigenvectors of `*A*` to construct P:`

v[1] = matrix([[1], [4/3], [0], [0]]), v[2] = matrix([[0], [0], [1], [4/3]]), v[3] = matrix([[-4/3], [1], [0], [0]]), v[4] = matrix([[0], [0], [-4/3], [1]])

P = matrix([[1, 0, -4/3, 0], [4/3, 0, 1, 0], [0, 1, 0, -4/3], [0, 4/3, 0, 1]]), P^`-1` = matrix([[9/25, 12/25, 0, 0], [0, 0, 9/25, 12/25], [-12/25, 9/25, 0, 0], [0, 0, -12/25, 9/25]])

P^`-1`*P*` = `*matrix([[9/25, 12/25, 0, 0], [0, 0, 9/25, 12/25], [-12/25, 9/25, 0, 0], [0, 0, -12/25, 9/25]])*matrix([[1, 0, -4/3, 0], [4/3, 0, 1, 0], [0, 1, 0, -4/3], [0, 4/3, 0, 1]]) = matrix([[1, 0,...

P^`-1`*A*P*` = `*matrix([[9/25, 12/25, 0, 0], [0, 0, 9/25, 12/25], [-12/25, 9/25, 0, 0], [0, 0, -12/25, 9/25]])*matrix([[-7, 24, 0, 0], [24, 7, 0, 0], [0, 0, -7, 24], [0, 0, 24, 7]])*matrix([[1, 0, -4/...

`A is diagonalizable using the eigenvectors of `*A*` to construct P:`

v[1] = matrix([[1], [4/3], [0], [0]]), v[2] = matrix([[0], [0], [1], [4/3]]), v[3] = matrix([[-4/3], [1], [0], [0]]), v[4] = matrix([[0], [0], [-4/3], [1]])

P = matrix([[1, 0, -4/3, 0], [4/3, 0, 1, 0], [0, 1, 0, -4/3], [0, 4/3, 0, 1]]), P^`-1` = matrix([[9/25, 12/25, 0, 0], [0, 0, 9/25, 12/25], [-12/25, 9/25, 0, 0], [0, 0, -12/25, 9/25]])

P^`-1`*P*` = `*matrix([[9/25, 12/25, 0, 0], [0, 0, 9/25, 12/25], [-12/25, 9/25, 0, 0], [0, 0, -12/25, 9/25]])*matrix([[1, 0, -4/3, 0], [4/3, 0, 1, 0], [0, 1, 0, -4/3], [0, 4/3, 0, 1]]) = matrix([[1, 0,...

P^`-1`*A*P*` = `*matrix([[9/25, 12/25, 0, 0], [0, 0, 9/25, 12/25], [-12/25, 9/25, 0, 0], [0, 0, -12/25, 9/25]])*matrix([[-7, 24, 0, 0], [24, 7, 0, 0], [0, 0, -7, 24], [0, 0, 24, 7]])*matrix([[1, 0, -4/...

``

`Now find an orthogonal matrix Q that orthogonally diagonalizes A.`

`Begin with the four eigenvectors of A and convert to an orthonormal set: `

v[1] = vector([1, 4/3, 0, 0])

v[2] = vector([0, 0, 1, 4/3])

v[3] = vector([-4/3, 1, 0, 0])

v[4] = vector([0, 0, -4/3, 1])

`Apply Gram-Schmidt Process to these four vectors: `

w[1] = v[1]

w[2] = v[2]-`(`*v[2]*` dot`*` w`[1]*`)`/` ||`/w[1]/`||`^2*`w `[1]

w[3] = v[3]-`(`*v[3]*` dot`*` w`[1]*`)`/` ||`/w[1]/`||`^2*`w `[1]-`(`*v[3]*` dot`*` w`[2]*`)`/` ||`/w[2]/`||`^2*`w `[2]

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]

w[1] = vector([1, 4/3, 0, 0])

w[2] = vector([0, 0, 1, 4/3])-`(`*vector([0, 0, 1, 4/3])*` dot`*vector([1, 4/3, 0, 0])*`)`/` ||`/vector([1, 4/3, 0, 0])/`||`^2*vector([1, 4/3, 0, 0])

w[3] = vector([-4/3, 1, 0, 0])-`(`*vector([-4/3, 1, 0, 0])*` dot`*vector([1, 4/3, 0, 0])*`)`/` ||`/vector([1, 4/3, 0, 0])/`||`^2*vector([1, 4/3, 0, 0])-`(`*vector([-4/3, 1, 0, 0])*` dot`*vector([0, 0, ...

w[4] = (vector([0, 0, -4/3, 1])-`(`*vector([0, 0, -4/3, 1])*` dot`*vector([1, 4/3, 0, 0])*`)`/` ||`/vector([1, 4/3, 0, 0])/`||`^2*vector([1, 4/3, 0, 0])-`(`*vector([0, 0, -4/3, 1])*` dot`*vector([0, 0,...
w[4] = (vector([0, 0, -4/3, 1])-`(`*vector([0, 0, -4/3, 1])*` dot`*vector([1, 4/3, 0, 0])*`)`/` ||`/vector([1, 4/3, 0, 0])/`||`^2*vector([1, 4/3, 0, 0])-`(`*vector([0, 0, -4/3, 1])*` dot`*vector([0, 0,...

`Obtain a basis of four orthogonal vectors: `

w[1] = vector([1, 4/3, 0, 0])

w[2] = vector([0, 0, 1, 4/3])

w[3] = vector([-4/3, 1, 0, 0])

w[4] = vector([0, 0, -4/3, 1])

`Obtain an orthonormal basis of four orthogonal unit vectors: `

u[1] = 1/` ||`/w[1]/`||`*` w`[1], `   ||`*w[1]*`||` = 5/3

u[2] = 1/` ||`/w[2]/`||`*` w`[2], `   ||`*w[2]*`||` = 5/3

u[3] = 1/` ||`/w[3]/`||`*` w`[3], `   ||`*w[3]*`||` = 5/3

u[4] = 1/` ||`/w[4]/`||`*` w`[4], `   ||`*w[4]*`||` = 5/3

u[1] = matrix([[3/5], [4/5], [0], [0]]), u[2] = matrix([[0], [0], [3/5], [4/5]]), u[3] = matrix([[-4/5], [3/5], [0], [0]]), u[4] = matrix([[0], [0], [-4/5], [3/5]])

`Use these orthonormal column vectors to construct Q:`

Q = matrix([[3/5, 0, -4/5, 0], [4/5, 0, 3/5, 0], [0, 3/5, 0, -4/5], [0, 4/5, 0, 3/5]]), Q^`-1` = matrix([[3/5, 4/5, 0, 0], [0, 0, 3/5, 4/5], [-4/5, 3/5, 0, 0], [0, 0, -4/5, 3/5]])

Q^`-1`*Q*` = `*matrix([[3/5, 4/5, 0, 0], [0, 0, 3/5, 4/5], [-4/5, 3/5, 0, 0], [0, 0, -4/5, 3/5]])*matrix([[3/5, 0, -4/5, 0], [4/5, 0, 3/5, 0], [0, 3/5, 0, -4/5], [0, 4/5, 0, 3/5]]) = matrix([[1, 0, 0, ...

Q^`-1`*A*Q*` = `*matrix([[3/5, 4/5, 0, 0], [0, 0, 3/5, 4/5], [-4/5, 3/5, 0, 0], [0, 0, -4/5, 3/5]])*matrix([[-7, 24, 0, 0], [24, 7, 0, 0], [0, 0, -7, 24], [0, 0, 24, 7]])*matrix([[3/5, 0, -4/5, 0], [4/...

`So, the symmetric matrix A is orthogonally diagonalizable using the orthogonal matrix Q.`