Abstract
Norms are used to measure the size of vector and matrices.
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Notes
- 1.
In the case of one vector norm \(\lVert \ \rVert \) on \({\mathbb {C}}^m\) and another vector norm \(\lVert \ \rVert _\beta \) on \({\mathbb {C}}^n\) we would define \(\lVert {\boldsymbol {A}}\rVert :=\max _{{\boldsymbol {x}}\ne 0}\frac {\lVert {\boldsymbol {A}}{\boldsymbol {x}}\rVert }{\lVert {\boldsymbol {x}}\rVert _\beta }\).
- 2.
Hint: Show that \(\lVert \tilde {{\boldsymbol {b}}}_1\rVert ^2_{\boldsymbol {A}}\cdots \lVert \tilde {{\boldsymbol {b}}}_n\rVert ^2_{\boldsymbol {A}}=\det ({\boldsymbol {A}})\).
- 3.
Hint: use Gershgorin’s theorem.
- 4.
Hint: We have 〈x, y〉 = s(x, y) + is(x, iy), where \(s({\boldsymbol {x}},{\boldsymbol {y}}):=\frac 14\big (\lVert {\boldsymbol {x}}+{\boldsymbol {y}}\rVert ^2-\lVert {\boldsymbol {x}}-{\boldsymbol {y}}\rVert ^2\big )\).
References
R.A. Horn, C.R. Johnson, Matrix Analysis, Second edition (Cambridge University Press, Cambridge, UK, 2013)
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Lyche, T. (2020). Matrix Norms and Perturbation Theory for Linear Systems. 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_8
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DOI: https://doi.org/10.1007/978-3-030-36468-7_8
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