Abstract
We have seen that a Hermitian matrix is positive definite if and only if it has positive eigenvalues. Eigenvalues and some related quantities called singular values occur in many branches of applied mathematics and are also needed for a deeper study of linear systems and least squares problems.
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Notes
- 1.
This can also be shown without using the Jordan factorization, see [9].
- 2.
Hint: show that there is a diagonal matrix D such that D −1AD is symmetric.
- 3.
Hint: use a suitable factorization of p and use c).
- 4.
Hint: Use Taylor’s theorem for the function f(t) = R(x − ty).
References
M.W. Hirsch, S. Smale, Differential Equations, Dynamical Systems, and Linear Algebra (Academic Press, San Diego, 1974)
R.A. Horn, C.R. Johnson, Topics in Matrix Analysis (Cambridge University Press, Cambridge, UK, 1991)
R.A. Horn, C.R. Johnson, Matrix Analysis, Second edition (Cambridge University Press, Cambridge, UK, 2013)
G.W. Stewart, Introduction to Matrix Computations (Academic press, New York, 1973)
G.W. Stewart, J. Sun, Matrix Perturbation Theory (Academic Press, San Diego, 1990)
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Lyche, T. (2020). Eigenpairs and Similarity Transformations. 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_6
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DOI: https://doi.org/10.1007/978-3-030-36468-7_6
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