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Determinants (a) are helpful if you want to invert matrices, (b) are easy to calculate for 2x2 and 3x3 matrices, (c) nicely generalise to the infinite-dimension

Determinants (a) are helpful if you want to invert matrices, (b) are easy to calculate for 2x2 and 3x3 matrices, (c) nicely generalise to the infinite-dimensional case were you take appropriate generalisations of sums and the totally antisymmetric tensor occurring in the definition of the determinant (d) are widely used in theoretical physics and hence should at least be included in first year undergrad classes on linear algebra in physics and maths.

stephencanon 13y ago
Determinants are sometimes useful for symbolic inversion of matrices, but are not used for numerical inversion of non-trivial matrices (for reasons of both performance and stability). It has been a long time since I studied functional analysis, and I was never a physicist of any sort, but my limited understanding is that there are two common generalizations for the infinite case, both of which handle the cases of most interest to physics but are not available for general operators, and which mostly agree up to a constant factor, or are at least capable of producing compatible results in some sense. I'm not sure I'd say they "nicely generalize", though I'd admit to being hasty in saying that they don't generalize. And yes, I agree that they're widely used in physics (which is why I think they should be taught to math/physics majors).
x0x0 13y ago
Determinants are never actually used to invert matrices. In fact, I can't think of any computational math that even calculates an inverse matrix. It's both faster, lower storage (unless you invert in place?) and far more numerically stable to solve Ax=b by factoring A as LU/chol/maybe QR. Even if you had multiple b you would just save your factorization. In which bit of math do people actually instantiate an inverse?
- stephencanon 13y ago
  > In which bit of math do people actually instantiate an inverse? In symbolic computations with very structured matrices, one sometimes constructs explicit inverses. This is an extreme corner case, and never really comes up except in some very simple physical models (in my experience).
- srean 13y ago
  > In which bit of math do people actually instantiate an inverse? I am assuming you are being rhetorical here. I also think your working hypothesis is that the only use of a matrix inverse is to solve a linear system of equations. Under this hypothesis it does make sense that there is no utility to computing inverses. However, this hypothesis is incorrect. There are tons of cases where you would want to get at the entries of an inverse, but has nothing to do with solving equations. Here area few examples: consider data generated from a multivariate Gaussians. Now you want to know which components are independent of each other. This is given by whether the inverse of the correlation matrix is zero or not. Another example, say you want to minimize a function defined over matrices, but want to minimize it only over the set of positive semi-definite matrices. One way of doing these is to use a barrier function: a function that blows up astronomically when you approach the boundary of your constraint set. For positive definite matrices such a barrier function is the logarithm of the determinant. I could go on...

stephencanon 13y ago

Determinants are sometimes useful for symbolic inversion of matrices, but are not used for numerical inversion of non-trivial matrices (for reasons of both performance and stability). It has been a long time since I studied functional analysis, and I was never a physicist of any sort, but my limited understanding is that there are two common generalizations for the infinite case, both of which handle the cases of most interest to physics but are not available for general operators, and which mostly agree up to a constant factor, or are at least capable of producing compatible results in some sense. I'm not sure I'd say they "nicely generalize", though I'd admit to being hasty in saying that they don't generalize. And yes, I agree that they're widely used in physics (which is why I think they should be taught to math/physics majors).

x0x0 13y ago

Determinants are never actually used to invert matrices. In fact, I can't think of any computational math that even calculates an inverse matrix. It's both faster, lower storage (unless you invert in place?) and far more numerically stable to solve Ax=b by factoring A as LU/chol/maybe QR. Even if you had multiple b you would just save your factorization. In which bit of math do people actually instantiate an inverse?

stephencanon 13y ago
> In which bit of math do people actually instantiate an inverse? In symbolic computations with very structured matrices, one sometimes constructs explicit inverses. This is an extreme corner case, and never really comes up except in some very simple physical models (in my experience).
srean 13y ago
> In which bit of math do people actually instantiate an inverse? I am assuming you are being rhetorical here. I also think your working hypothesis is that the only use of a matrix inverse is to solve a linear system of equations. Under this hypothesis it does make sense that there is no utility to computing inverses. However, this hypothesis is incorrect. There are tons of cases where you would want to get at the entries of an inverse, but has nothing to do with solving equations. Here area few examples: consider data generated from a multivariate Gaussians. Now you want to know which components are independent of each other. This is given by whether the inverse of the correlation matrix is zero or not. Another example, say you want to minimize a function defined over matrices, but want to minimize it only over the set of positive semi-definite matrices. One way of doing these is to use a barrier function: a function that blows up astronomically when you approach the boundary of your constraint set. For positive definite matrices such a barrier function is the logarithm of the determinant. I could go on...