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stephenmalina.com
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| | | | | Selected Exercises # 5.A # 12. Define $ T \in \mathcal L(\mathcal P_4(\mathbf{R})) $ by $$ (Tp)(x) = xp'(x) $$ for all $ x \in \mathbf{R} $. Find all eigenvalues and eigenvectors of $ T $. Observe that, if $ p = a_0 + a_1 x + a_2 x^2 + a_3 x^3 + a_4 x^4 $, then $$ x p'(x) = a_1 x + 2 a_2 x^2 + 3 a_3 x^3 + 4 a_4 x^4. | |
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www.reedbeta.com
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| | | | | Pixels and polygons and shaders, oh my! | |
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nhigham.com
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| | | | | The numerical range of a matrix $latex A\in\mathbb{C}^{n\times n}$, also known as the field of values, is the set of complex numbers $latex W(A) = \biggl\{\, \displaystyle\frac{z^*Az}{z^*z}: 0\ne z\in\mathbb{C}^n \,\biggr\}. $ The set $LATEX W(A)$ is compact and convex (a nontrivial property proved by Toeplitz and Hausdorff), and it contains all the eigenvalues of $LATEX... | |
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thatsmaths.com
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| | | In last week's post, we defined an extension of parity from the integers to the rational numbers. Three parity classes were found --- even, odd and none. This week, we show that, with an appropriate ordering or enumeration of the rationals, the three classes are not only equinumerate (having the same cardinality) but of equal... | ||
