Question

Consider the matrix \( A = \begin{pmatrix} 9 & 15 \\ 15 & 50 \end{pmatrix} \). The matrix A is decomposed using Cholesky decomposition. Determine the value of \(l_{22}\).

Hint:

For a 2x2 matrix \(A = \begin{pmatrix} a & b
b & c \end{pmatrix}\), the Cholesky elements can be found directly: \(l_{11} = \sqrt{a}\), \(l_{21} = b/l_{11}\), and \(l_{22} = \sqrt{c - l_{21}^2}\).

Answer 1

Step 1: Decomposition Setup.
$A = LL^T = \begin{pmatrix} l_{11} & 0 \\ l_{21} & l_{22} \end{pmatrix} \begin{pmatrix} l_{11} & l_{21} \\ 0 & l_{22} \end{pmatrix} = \begin{pmatrix} l_{11}^2 & l_{11}l_{21} \\ l_{11}l_{21} & l_{21}^2 + l_{22}^2 \end{pmatrix}$.
Step 2: Equate Elements.
1) $l_{11}^2 = 9 \implies l_{11} = 3$. 2) $l_{11}l_{21} = 15 \implies 3 \times l_{21} = 15 \implies l_{21} = 5$. 3) $l_{21}^2 + l_{22}^2 = 50$.
Step 3: Solve for $l_{22$.}
$5^2 + l_{22}^2 = 50 \implies 25 + l_{22}^2 = 50 \implies l_{22}^2 = 25 \implies l_{22} = 5$.