(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Most invertible matrices can be written as a product A=LU of a lower triangular matrix L and an upper triangular matrix U, where in addition all diagonal entries of U are 1.

a. Prove uniqueness, that is, prove that there is at most one way to write A as a product.

b. Explain how to compute L and U when the matrix A is given.

c. Show that every invertible matrix can be written as a product LPU, where L, U, are as above and P is a permutation matrix.

2. Relevant equations

\\

3. The attempt at a solution

b. The general process involves multiplying by a series of lower triangular matrices to "kill" entries under the diagonal of A. Eventually, A will be in upper triangular form. We will have

[tex]U = L_k...L_1A \Rightarrow A = LU[/tex] where [tex]L = L_1^{-1}...L_k^{-1}[/tex]

Notes: The product of lower triangular matrices is also triangular. Also, lower triangular matrices are invertible.

I'm stuck on parts a. and c.

Thanks in advance for any help!

**Physics Forums | Science Articles, Homework Help, Discussion**

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Homework Help: LU factorization of matrices

**Physics Forums | Science Articles, Homework Help, Discussion**