LLT-Factorization: Lower Triangular Matrix with Positive Diagonal for 4x3 Matrix

[DeadxBunny]

Homework Statement

Determine directly the LLT-Factorization, in which L is a lower triangular matrix with positive diagonal elements, for the matrix

| 4 1/2 1 |
| 1/2 17/16 1/4 |
| 1 1/4 33/64|

Homework Equations

I don't know.

The Attempt at a Solution

I don't know what to do; please help!

Thanks!

 

[AKG]

You mean LL^T, where L^T is the transpose of L? This is easy, why don't you just do it? Let L be an arbitrary lower triangular matrix, and compute the entries of LL^T in terms of the entries of L. Then equate them with the corresponding entries in the matrix you're given to work with, and solve the equations. For example, if you had a 1x1 matrix (7), and were asked to find the LL^T factorization, you let L = (a) be an arbitrary lower triangular matrix. Then you can compute the entries of LL^T in terms of the entries of L: LL^T = (a)(a) = (a²). Then you equate entries of this matrix with the entries of the matrix you're working with, (7). So you get a² = 7. So a is either the positive or negative square root of 7, but since we have the condition "L has positive elements on its diagonal" we know a to be the positive root of 7. Your problem is similar, it's just that you'll get 3 + 2 + 1 = 6 equations instead of just 1 equation.