Answer by AccidentalTaylorExpansion for Do matrices really rotate and stretch...
You are quite close. One can always decompose a matrix using the singular value decomposition (SVD). For some matrix $M$ this is written as$$M=U\Sigma V^*,$$where $V^*$ means the conjugate transpose of...
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Answer by Stéphane Jaouen for Do matrices really rotate and stretch vectors...
Consider the matrix $A=\begin{bmatrix}0 & -1 \\1 & 0\end{bmatrix}$. In the Euclidean plane, this matrix is the matrix in the canonical basis of rotation with $+\frac{pi}2$. For example,...
View ArticleAnswer by mweiss for Do matrices really rotate and stretch vectors or is that...
When you ask what a matrix "really is", you are asking the wrong question. A matrix is, at its heart, just a big block of numbers arranged in a rectangular grid. That's it.The important question is:...
View ArticleAnswer by Lorenzo Pompili for Do matrices really rotate and stretch vectors...
First of all: "stretching and rotating" is not a definition, is intuition. The intuition on stretching and rotation you have is ok, although it depends a little on what you mean. All functions rotate...
View ArticleAnswer by pancini for Do matrices really rotate and stretch vectors or is...
I don't know what your friend was getting at, but a (square) matrix—meaning an $n\times n$ array of elements in some field $k$—can always be interpreted as representing a linear transformation on...
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Answer by AbysmalMathematics for Do matrices really rotate and stretch...
In the “linear transformation” point of view, matrices represent linear transformations, by denoting where the unit vector in the $x$ direction goes in the first column and where the unit vector in the...
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Answer by ronno for Do matrices really rotate and stretch vectors or is that...
I'm not sure what distinction the word locally is trying to convey. Perhaps your friend was talking about matrix fields, ie continuously varying matrices? Though that doesn't seem particularly relevant...
View ArticleDo matrices really rotate and stretch vectors or is that definition incorrect?
I come from the applied math and statistics world, but I was talking to my friend who comes from the pure math and number theory world—in particular Galois representation theory.We got together to...
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