Question

Prove the following statement: If a, b, c, and d satisfy a > 0, d > 0, ad > bc, then the reduced non-negative factorization is: Consider the uniqueness of the nonnegative Cholesky factorization (Theorem 3.5.15) of matrix A, including the explanation of why it exists for A. Substitute the factorization into A and compare it to the Cholesky factorization.

Answer 1

To prove the given statement, we can consider the nonnegative Cholesky factorization of matrix A and substitute it into the equation. By comparing it with the Cholesky factorization, we can find the reduced non-negative factorization.

Step-by-step explanation:

In order to prove the given statement, let's consider the nonnegative Cholesky factorization of a matrix A. The Cholesky factorization is a way of expressing a positive definite matrix A as the product of a lower triangular matrix L and its transpose. The uniqueness of the nonnegative Cholesky factorization guarantees that there is only one factorization of A into L and L_transpose.

To substitute the nonnegative Cholesky factorization into A, we can rewrite A as L * L_transpose. By comparing this with the Cholesky factorization, we can see that the reduced non-negative factorization is given by L. This guarantees that the nonnegative Cholesky factorization exists for matrix A and provides the desired factorization.