[Numpy-discussion] determinant of a scalar not handled

[Robert Kern]

On Tue, Jul 27, 2010 at 07:51, Skipper Seabold <jsseabold at gmail.com> wrote:
> On Mon, Jul 26, 2010 at 10:05 PM, Alan G Isaac <aisaac at american.edu> wrote:
>> But I am still confused about the use case.
>> What is the scalar- (or 1d-array-) returning procedure
>> invokedbefore taking the determinant?
>
> Recently I ran into this trying to make the log-likelihood of a
> multivariate and univariate autoregressive process use the same
> function.  One has log(sigma_scalar) and one calls for
> logdet(Sigma_matrix).  I also ran in to again yesterday working on the
> Kalman filter, depending on the process being modeled and how the user
> writes a function if the needed coefficient arrays depend on
> parameters.  To be more general, I have to put in atleast_2d, even
> though these checks are really in slogdet.

Not necessarily. In this context, you are treating a scalar as a 1x1
matrix. Or rather, in full generality, you have a 1x1 matrix and
1-element vectors and only doing operations on them that map fairly
neatly onto a subset of scalar properties. Consequently, you can use
scalars in their place without much problem (to add confusion, the
scalar case was formulated first, then generalized to the multivariate
case, but that doesn't change the mathematics unless if you believe
certain ethnomathematicians).

However, there are other contexts in which scalars are used where the
determinant would come into play. For example, scalar-vector
multiplication is defined. If you have an n-vector, then scalar-vector
multiplication behaves like matrix-vector multiplication provided that
the matrix is a diagonal matrix with the diagonal entries each being
the scalar value. In this context, the determinant is not just the
scalar value itself, but rather value**n.

Many of the references you found stating that the "determinant of a
scalar value is" the scalar itself were actually referring to 1x1
matrices, not true scalars. 1x1 matrices behave like scalars, but not
all scalars behave like 1x1 matrices. linalg.det() does not know the
context in which you are treating the scalar, so it rightly complains.

That said, I expect you will be running into this 1x1<->scalar special
case reasonably frequently in statsmodels. Writing a dwim_logdet()
utility function there that does what you want is a perfectly
reasonable thing to do.

-- 
Robert Kern

"I have come to believe that the whole world is an enigma, a harmless
enigma that is made terrible by our own mad attempt to interpret it as
though it had an underlying truth."
  -- Umberto Eco