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g-w1.github.io | ||
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nulliq.dev
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| | | | | In search of a better dot product | |
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arkadiusz-jadczyk.eu
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| | | | | In the last post, Geodesics of left invariant metrics on matrix Lie groups - Part 1,we have derived Arnold's equation - that is a half of the problem of finding geodesics on a Lie group endowed with left-invariant metric. Suppose $G$ is a Lie group, and $g(\xi,\eta)$ is a scalar product (i.e. | |
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nhigham.com
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| | | | | The trace of an $latex n\times n$ matrix is the sum of its diagonal elements: $latex \mathrm{trace}(A) = \sum_{i=1}^n a_{ii}$. The trace is linear, that is, $latex \mathrm{trace}(A+B) = \mathrm{trace}(A) + \mathrm{trace}(B)$, and $latex \mathrm{trace}(A) = \mathrm{trace}(A^T)$. A key fact is that the trace is also the sum of the eigenvalues. The proof is by... | |
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nhigham.com
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| | | A norm on $latex \mathbb{C}^{m \times n}$ is unitarily invariant if $LATEX \|UAV\| = \|A\|$ for all unitary $latex U\in\mathbb{C}^{m \times m}$ and $latex V\in\mathbb{C}^{n\times n}$ and for all $latex A\in\mathbb{C}^{m \times n}$. One can restrict the definition to real matrices, though the term unitarily invariant is still typically used. Two widely used matrix norms... | ||
