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nickhar.wordpress.com | ||
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nhigham.com
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... |
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www.ethanepperly.com
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theoryofcomputing.org
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www.jeremykun.com
Last time we worked through some basic examples of universal properties, specifically singling out quotients, products, and coproducts. There are many many more universal properties that we will mention as we encounter them, but there is one crucial topic in category theory that we have only hinted at: functoriality. As we've repeatedly stressed, the meat of category theory is in the morphisms. One natural question one might ask is, what notion of morphism is there between categories themselves? |
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