# LU factorization of matrices

## Homework Statement

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.

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## 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

$$U = L_k...L_1A \Rightarrow A = LU$$ where $$L = L_1^{-1}...L_k^{-1}$$

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!

Office_Shredder
Staff Emeritus