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Re: Why does Transpose[m, {1, 1}] give the diagonal of m?
*To*: mathgroup at christensen.cybernetics.net
*Subject*: [mg1886] Re: [mg1859] Why does Transpose[m, {1, 1}] give the diagonal of m?
*From*: Richard Mercer <richard at seuss.math.wright.edu>
*Date*: Sat, 12 Aug 1995 22:51:00 -0400
> An elegant test. My question: why does Transpose[m, {1,
> 1}] return the list of diagonal elements? Looking under
> Transpose in Wolfram's Mathematica Reference Guide (an
> appendix to Mathematica, 2nd ed.), I find:
> Transpose[list, {n1,n2, ...}] transposes list so that the kth level
> in list is the nkth level in the result.
> I expected Depth[Transpose[list, {n1,n2, ...}]] to be
> the same as Depth[list]. I expected Transpose to swap
> levels; I did not expect it to throw any away. Evidently
> I am laboring under some misconception. Can someone
> explain what is going on?
I think so, at the price of a certain amount of abstraction and background
discussion.
Consider the matrix
A = {{16,27},{38,49}}, or
A = [ 16 27 ]
[ 38 49 ]
Then
C = Transpose[A,{1,1}] gives
{16,49}
Why?
*Short Version*
Since {1,1} is not a permutation, Transpose[list,{1,1}] is not defined by the
Reference Guide. The effect of using {1,1} is to say the two entries must be
the same, so C[[1]] = A[[1,1]] = 16, and C[[2]] = A[[2,2]] = 49. The effect is
to compress a two-dimensional list into a one-dimensional list.
If you believe that, I have some Netscape shares I'd like to sell you!
So you'd better read the...
*Long Version*
(0) The discussion below works just as well for non-square matrices and their
higher-dimensional analogues, but it's simplest to consider a 2x2 matrix.
(1) The structured list objects on which Transpose operates can be viewed as
functions on certain integer lattices. Sounds fancy, but an example shows it's
really a simple idea.
The matrix A can be thought of as a function of two indices with the following
values:
A(1,1) = 16
A(1,2) = 27
A(2,1) = 38
A(2,2) = 49
(In Mathematica you use double brackets instead of parentheses.)
The domain of the function A is therefore the set of ordered pairs
D = {(1,1), (1,2), (2,1), (2,2)}.
(set notation, not Mathematica notation!)
D is the same as the set (cartesian) product {1,2} X {1,2}.
(2) The ordinary transpose operation can be viewed as the composition of the
function A and the map t defined from D to D as follows:
t(1,1) = (1,1)
t(1,2) = (2,1)
t(2,1) = (1,2)
t(2,2) = (2,2)
So if B = Transpose[A], then
B(1,1) = A(t(1,1)) = A(1,1) = 16
B(1,2) = A(t(1,2)) = A(2,1) = 38
B(2,1) = B(t(2,1)) = A(1,2) = 27
B(2,2) = B(t(2,2)) = A(2,2) = 49
Therefore B = {{16,38},{27,49}} in Mathematica notation, or
B = [ 16 38 ]
[ 27 49 ]
as expected.
(3) The key is to understand the map t and how it can be generalized to other
situations. The map t is defined by a "permutation" or reordering applied to
the elements of the domain D. In this case the reordering is described by {2,1}
because
-- the first entry of the result is the second(2) entry of the input,
-- the second entry of the result is the first(1) entry of the input, i.e.
t(a,b) = (b,a). Therefore
Transpose[A,{2,1}] means the same as Transpose[A], and
Transpose[A,{1,2}] = A,
because the permutation {1,2} means no reordering at all.
(4) If we now extend this discussion to the non-permutation {1,1}, it should
define a map s by
-- the first entry of the result is the first(1) entry of the input,
-- the second entry of the result is the first(1) entry of the input,
i.e. s(a) = (a,a).
Why not s(a,b) = (a,a)? Since only the first entry of the input is used, we
assume the input has only one entry.
s maps from {1,2} into {1,2} X {1,2} = D.
(5) By analogy, C = Transpose[A,{1,1}] should now represent the composition of
the function A with the map s. The result is a function on {1,2}, with values
C(1) = A(s(1)) = A(1,1) = 16
C(2) = A(s(2)) = A(2,2) = 49
This discussion could be continued to cases of higher dimensional lists, but
I'm amazed if you've read this far!
Exercise for the reader:
If you have a three dimension list, e.g.
A = {{{16,27},{38,49}},{{91,82},{73,64}}}
what should be the result of Transpose[A,{1,1,2}] ?
Explain why Mathematica accepts the non-permutation {1,1,2} but not {1,1,3} in
its place.
Richard Mercer
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