Abstract
In this chapter we consider LU factorizations of Hermitian and positive definite matrices. Recall that a matrix \({\boldsymbol {A}}\in {\mathbb {C}}^{n\times n}\) is Hermitian if A āā=āA, i.e., \(a_{ji}=\overline {a}_{ij}\) for all i, j. A real Hermitian matrix is symmetric. Since \(a_{ii}=\overline {a}_{ii}\) the diagonal elements of a Hermitian matrix must be real.
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Notes
- 1.
Hint: The matrix E ā1 is of the form E ā1ā=āIā+āauu T for some \(a \in {\mathbb {R}}\).
References
G.H. Golub, C.F. Van Loan, Matrix Computations, 4th Edition (The John Hopkins University Press, Baltimore, MD, 2013)
C.D. Meyer, Matrix Analysis and Applied Linear Algebra. (SIAM, Philadelphia, 2000)
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Lyche, T. (2020). LDL* Factorization and Positive Definite Matrices. In: Numerical Linear Algebra and Matrix Factorizations. Texts in Computational Science and Engineering, vol 22. Springer, Cham. https://doi.org/10.1007/978-3-030-36468-7_4
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DOI: https://doi.org/10.1007/978-3-030-36468-7_4
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