Re: Determinants of multi-dimensional matrices
- From: lrudolph@xxxxxxxxx (Lee Rudolph)
- Date: 1 Dec 2005 19:57:36 -0500
"Tero" <tero13@xxxxxxxxx> writes:
>Hi there. I study Electrical Engineering (first year), and I am being
>bothered by by Linear Algebra issues lately. One of them is the
>three-(or more)-dimensional matrices. In Linear Algebra lessons, we
>only have to deal with two-dimensional matrices, but, in fact, there is
>no limitation in the number of dimensions, it could be anything. Yet,
>all definitions we have learnt are designed for two-dimensional
>matrices. One of them is the determinant. I am trying to discover on my
>own, how to compute the determinant of a NxNxN matrix, but I am
>baffled. Maybe I should ask if there IS such a thing as determinant of
>a three-dimensional matrix.
>
>Another one is the multiplication of three-dimensional matrices. All
>clear on how to multiply an MxN with an NxK matrix. But what about
>three dimensional matrices? No clue at all...
You are, apparently, being taught "Matrix Algebra", as a kind
of applied algebra (presumably tailored to EE applications).
Although it is not necessary to do so, it is *possible* to
teach "Matrix Algebra" in a way that leaves both matrix
multiplication and determinants nearly completely unmotivated,
and presents them simply in terms of (mysterious) formulas.
There are other ways to approach "Linear Algebra", some algebraic,
some geometric, in which it's much more difficult to avoid an
early introduction of motivation for matrix multiplication,
and in which it is at least possible (though it can be difficult,
depending on the nature of the students) to introduce motivation
for determinants more or less at the same time formulas for
determinants are introduced.
When multiplication and determinants are introduced purely
in terms of formulas, it is *extremely* natural (and reasonable)
for a student such as yourself (or me, though I was not yet an
enrolled student of mathematics, merely a kid visiting a friend's
uncle who was an engineer, and trying to make sense of some books
on the shelves) to wonder how to multiply "higher dimensional"
matrices, and how to evaluate their "determinants". Unfortunately,
there is no particularly reasonable way to do either of those
things. Once you learn the motivation for matrix multiplication
(in terms of "linear transformations" and their compositions)
and determinants, it's much easier to see why the case of "two
dimensional" matrices is the one where matrix multiplication and
determinants make sense.
Lee Rudolph
.
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