Solving a Linear System using Cholesky method Confusion

Question

Hello I am trying to solve the following system using the cholesky method in Matlab

A = [1 -1; -1 5];

L = [ 1 0;

-1 2]; //Lower Triangular Decomp via choleksy method

Lt = [1 -1; 0 2]; //L transpose

B = [-1; 9]; //B vector since were solving Ax = B -> L*Lt*x = B

Linv = inv(L) //calculating inverse of L

B*Linv //Computing multiplication

Now... I dont understand why I get the error

Error using *
Incorrect dimensions for matrix multiplication. Check that the number of columns in the first matrix matches the number of rows in the second matrix.

I am guessing its because Im multyplying in this order? 2 x 1 * 2 x 2 so the dimensions dont match up. Maybe I dont understand how this process works fully, but I thought in order to solve the system I would have to eliminate L by multiplying both sides by L inverse.

Answer 1

You are doing B*Linv but what you want is Linv*B.

This is because if $LL^{T}\mathbf{x} = \mathbf{b}$, then we can multiply both sides on the left by $L^{-1}$ to obtain

$$L^{-1}LL^{T}\mathbf{x} = L^{-1}\mathbf{b}\Rightarrow L^{T}\mathbf{x} = L^{-1}\mathbf{b}.$$

(Remember, for matrix multiplication, $AB$ is not the same thing as $BA$ in general. Also in this case, $\mathbf{b}L^{-1}$ would not make sense because in a matrix product like $AB$, the number of columns in the first matrix ($A$) must match the number of rows in the second matrix ($B$).)