       Re: The Case of the Mystery Option

• To: mathgroup at christensen.cybernetics.net
• Subject: [mg1086] Re: The Case of the Mystery Option
• From: Richard Mercer <richard at seuss.math.wright.edu>
• Date: Sun, 14 May 1995 20:21:28 -0400

```>  Hi Group;
>   In Roman Maeder's excellent book Programming in
>  Mathematica, on page 100 is an example using the option
>  to Transpose.  (Up until this moment, I didn't even know
>  there was an option with Transpose, but never mind this.)
>  In essence,  Transpose[mat,{1,1}] returns a list of the
>  diagonal entries of the matrix  mat.  With the exception
>  of this one example, I have never seen an illustration
>  of the use of this option.  So I ask you kind souls to
>  answer a few questions for me:
>

>  (1)  Is it true that the 2nd line on page 132 of The Book
>  is a misprint?  It says that  Transpose[list,n]
>  interchanges the top level with the nth level.
>

>  (2)  From the error messages I have received while
>  experimenting with this option, I have deduced that the
>  option is a permutation.  It permutes levels?  Is this
>  correct?
>

>  (3)  How does one explain Maeder's example?
>

>  (4)  I guess that besides the Maeder example, the only
>  other uses appears in Tensors?
>

>    As usual, your help is more than appreciated.
>

>  Jack
>

Jack,
This "option" to Transpose has many other uses for displaying
multidimensional data tables besides mathematical tensors, though it always
involves "tensors" in the sense of lists satisfying TensorQ. Below is a visual
example of a three-dimensional list display in original form, transpose form,
then using all six permutations. Briefly,
(1) the identity permutation {1,2,3} has no effect (surprise)
(2) the permutation {2,1,3} has the same effect as plain Transpose:
it transposes the large 2x2 "block matrix"
(3) the permutation {1,3,2} has the effect of transposing each of
the "upper" (A) and "lower" (B) block matrices separately
(4) the permutations {2,3,1} and {3,1,2} have the effects, respectively,
of (3) followed by (2), and of (2) followed by (3)
(I think! This is making me dizzy...)
(5) the permutation {3,2,1} "shuffles" the original "upper" (A) and "lower" (B)

block matrices, pairing together matching entries of these blocks.

Richard Mercer

threedeep = Outer[List,{A,B},{1,2},{i,ii}]
{{{{A, 1, i}, {A, 1, ii}}, {{A, 2, i}, {A, 2, ii}}},

{{{B, 1, i}, {B, 1, ii}}, {{B, 2, i}, {B, 2, ii}}}}

TableForm[threedeep]
A 1 i    A 2 i
A 1 ii   A 2 ii

B 1 i    B 2 i
B 1 ii   B 2 ii

TableForm[Transpose[threedeep]]
A 1 i    B 1 i
A 1 ii   B 1 ii

A 2 i    B 2 i
A 2 ii   B 2 ii

TableForm[Transpose[threedeep,{1,2,3}]]
A 1 i    A 2 i
A 1 ii   A 2 ii

B 1 i    B 2 i
B 1 ii   B 2 ii

TableForm[Transpose[threedeep,{1,3,2}]]
A 1 i   A 1 ii
A 2 i   A 2 ii

B 1 i   B 1 ii
B 2 i   B 2 ii

TableForm[Transpose[threedeep,{2,1,3}]]
A 1 i    B 1 i
A 1 ii   B 1 ii

A 2 i    B 2 i
A 2 ii   B 2 ii

TableForm[Transpose[threedeep,{2,3,1}]]
A 1 i    B 1 i
A 2 i    B 2 i

A 1 ii   B 1 ii
A 2 ii   B 2 ii

TableForm[Transpose[threedeep,{3,1,2}]]
A 1 i   A 1 ii
B 1 i   B 1 ii

A 2 i   A 2 ii
B 2 i   B 2 ii

TableForm[Transpose[threedeep,{3,2,1}]]
A 1 i    A 2 i
B 1 i    B 2 i

A 1 ii   A 2 ii
B 1 ii   B 2 ii

```

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