Exercise · ZW-CHOLESKY-FACTORIZATION-006 · ★★☆ Intermediate
Compute \(l_{11}\) and \(l_{21}\) for \(A=\begin{pmatrix}4&2\\2&5\end{pmatrix}\).
Choose one option.
Show answer
- A. \(l_{11}=2\), \(l_{21}=1\)
Explanation
We have \(l_{11}=\sqrt{4}=2\) and \(l_{21}=2/2=1\). The option that leaves \(l_{11}\) without the root and the one that uses \(a_{11}\) directly are common 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-007 If a_22=10 and l_21=3, what is l_22?
- ZW-CHOLESKY-FACTORIZATION-008 For A=[9 3; 3 5], what is the factor L?
- ZW-CHOLESKY-FACTORIZATION-009 If L=[2 0; 1 2], what is the product LL^T?
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