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Results tagged with linear-algebra
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user 428
Questions about the properties of vector spaces and linear transformations, including linear systems in general.
3
votes
Accepted
factorization of the product of a matrix element and its cofactor
For the main problem, we can use expansion by minors along the ith row to compute
$\det(A_n) = (-1)^{i+1}(a(n)_{i1}\widetilde{a(n)_{i1}} - a(n)_{i2}\widetilde{a(n)_{i2}} + a(n)_{i3}\widetilde{a(n)_{i …
- 6,589
4
votes
Accepted
Conjugation in SU(2)
It's not hard to explicitly construct G using the quaternions, assuming P is not \pm Q, and I think this is worth working out in detail because I really like this picture of SU(2). Identify SU(2) wit …
- 6,589
4
votes
Accepted
Lifting bases for (Z/pZ)^n to Z^n
Here's a unified argument based on my comments to Scott's post that doesn't use quadratic reciprocity in any form. Suppose n=2 and p >= 5, and lift each line of slope i in Y(2,p) to a point (ai+pbi, …
- 6,589
15
votes
A random walk matrix has eigenvalue 1 with multiplicty 1 - why?
For large enough n, the matrix B = A + A2 + ... + An has positive entries since there's a path of length at most n connecting any two vertices. Thus by the Perron-Frobenius theorem B has a unique max …
- 6,589