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pfzhang.wordpress.com | ||
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nhigham.com
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| | | | | For a polynomial $latex \notag \phi(t) = a_kt^k + \cdots + a_1t + a_0, $ where $latex a_k\in\mathbb{C}$ for all $latex k$, the matrix polynomial obtained by evaluating $latex \phi$ at $latex A\in\mathbb{C}^{n \times n}$ is $latex \notag \phi(A) = a_kA^k + \cdots + a_1A + a_0 I. $ (Note that the constant term is... | |
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mikespivey.wordpress.com
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| | | | | The Riemann zeta function $latex \zeta(s)$ can be expressed as $latex \zeta(s) = \sum_{n=1}^{\infty} \frac{1}{n^s}$, for complex numbers s whose real part is greater than 1. By analytic continuation, $latex \zeta(s)$ can be extended to all complex numbers except where $latex s = 1$. The power sum $latex S_a(M)$ is given by $latex S_a(M) =... | |
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nhigham.com
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| | | | | The Cayley-Hamilton Theorem says that a square matrix $LATEX A$ satisfies its characteristic equation, that is $latex p(A) = 0$ where $latex p(t) = \det(tI-A)$ is the characteristic polynomial. This statement is not simply the substitution ``$latex p(A) = \det(A - A) = 0$'', which is not valid since $latex t$ must remain a scalar... | |
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chava61photography.photo.blog
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