Fast Polar Decomposition Of A sTrAnGe Square Matrix - Help

[emlio]

Fast Polar Decomposition Of A sTrAnGe Square Matrix --!Help!

Homework Statement

| 2 0 4 |
| 0 1 0 | = A
| 3 0 2 |

Calculate the Polar Decomposition of A: $$A = LP$$

$$L =^{t}L > 0$$ L is symmetric and positive defined

$$P^{t}P = I$$

$$^{t}P=P^{-1}$$ P is an isometry: Transpose of P is egual to P^-1


Is there a quick way to do this?
I know how to do it but is very slow, and I think that must exists a quick way or some tricks to resolve the exercise, infact as you can see the matrix is very particular...

Help Me please...Tnx

 

[lippyka]

Homework EquationsA = LP L =^{t}L > 0 L is symmetric and positive definedP^{t}P = I ^{t}P=P^{-1} P is an isometry: Transpose of P is egual to P^-1The Attempt at a SolutionYou can calculate the polar decomposition of A using the Singular Value Decomposition (SVD). The SVD of A is A = U diag(σ) V^T, where U and V are orthogonal matrices and σ is a vector of singular values. Then the polar decomposition of A is A = U diag(σ) V^T = (U diag(σ^(1/2)))(V diag(σ^(1/2))). The first factor is the symmetric, positive definite square root of A, and the second is the isometry that maps A to its square root.