I move my arm. [holonomy] I move my arm, hand and shoulder. [holonomy] I move my arm, hand, shoulders, but my fingers are still. I move my arm. [lie group] I move my arm across the table. [embedded in a … Continue reading
multiplicitiesoffreedom: Gotcha.
@IgorCarron blogs recent applications of compressive sensing and matrix factorisation every week. (Compressive sensing solves underdetermined systems of equations, for example trying to fill in missing data, by L₁-norm minimisation.) This week: reverse-engineering biochemical pathways and complex systems analysis. (Source: … Continue reading
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Once you’re comfortable with 2-arrays and 2-matrices, you can move up a dimension or two, to 4-arrays or 4-tensors. You can move up to a 3-array / 3-tensor just by imagining a matrix which “extends back into the blackboard”. Like … Continue reading
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Linear Transformations will take you on a Trip Comparable to that of Magical Mushroom Sauce, And Perhaps cause More Lasting Damage Long after I was supposed to “get it”, I finally came to understand matrices by looking at the above … Continue reading
Mathematical matrices are blocks of numbers. So they don’t necessarily have definite characteristics the way plain vanilla, singleton numbers do. They’re freer. For example 131 ∈ positive and −9 ∈ negative. But is positive, negative, or zero? Well, it has … Continue reading
This is for my homies in math class. Mathematical matrices are blocks of numbers, arrayed in 2-D. (Higher-dimensional arrays are called tensors.) Left “times” right equals target. Each entry in the target is the result of a series of +’s … Continue reading
A matrix ℳ represents a sequence of + and × operations. At the end you’ve linearly transformed a space (sheared it, expanded it, rotated it — but kept the origin where it is.) Did the amount of stuff in the picture change when you … Continue reading
The eigenvectors of a matrix summarise what it does. Think about a large, not-sparse matrix. A lot of computations are implied in that block of numbers. Some of those computations might overlap each other—2 steps forward, 1 step back, 3 steps … Continue reading
Human Mathematics 