Exercise · ZW-CHOLESKY-FACTORIZATION-008 · ★★☆ Intermediate
For \(A=\begin{pmatrix}9&3\\3&5\end{pmatrix}\), what is the factor \(L\)?
Choose one option.
Show answer
- C. \(\begin{pmatrix}3&0\\1&2\end{pmatrix}\)
Explanation
We have \(l_{11}=3\), \(l_{21}=3/3=1\), and \(l_{22}=\sqrt{5-1}=2\). The option with entries copied directly from \(A\) and the upper triangular option are typical errors.
Review the concept: Cholesky factorization
The Cholesky factorization decomposes a symmetric positive definite matrix \(A\) as \(A = LL^{T}\), where \(L\) is lower triangular with positive diagonal.
\[A = LL^{T}, \; l_{jj} = \sqrt{a_{jj} - \sum_{k
More practice problems: cholesky factorization
- ZW-CHOLESKY-FACTORIZATION-001 What property must a matrix A have to apply the Cholesky factorization?
- ZW-CHOLESKY-FACTORIZATION-002 What form does the factor L have in A = LL^T?
- ZW-CHOLESKY-FACTORIZATION-003 If L is the Cholesky factor, what does the equality A = LL^T express?
- ZW-CHOLESKY-FACTORIZATION-004 What is the element l_11 of the Cholesky factor when a_11=9?
- ZW-CHOLESKY-FACTORIZATION-005 What is the relation between Cholesky and LU for a symmetric positive definite matrix?
- ZW-CHOLESKY-FACTORIZATION-006 Compute l_11 and l_21 for A=[4 2; 2 5].
- ZW-CHOLESKY-FACTORIZATION-007 If a_22=10 and l_21=3, what is l_22?
- ZW-CHOLESKY-FACTORIZATION-009 If L=[2 0; 1 2], what is the product LL^T?
← ZW-CHOLESKY-FACTORIZATION-007 ZW-CHOLESKY-FACTORIZATION-009 →