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Transpose, Re: The Case of the Mystery Option
*To*: mathgroup at christensen.cybernetics.net
*Subject*: [mg1131] Transpose, Re: [mg1056] The Case of the Mystery Option
*From*: Allan Hayes <hay at haystack.demon.co.uk>
*Date*: Wed, 17 May 1995 02:39:13 -0400
>From: Jack Goldberg <jackgold at math.lsa.umich.edu> in [mg1056] The
Case of the Mystery Option asks some questions about transpose.
Robert Villegas responded in mg[1063].
Jack asked about uses of the second parameter in Transpose other
than for getting the diagonal of a square matrix by
Transpose[M,{1,1}]. I did find a use for calculating the determinant
(mathgroup April 1992 I think). The result is given after the
following notes.
********
First, let's look at the way we extract the elements of a tensor T.
Suppose that its tensor rank is 4 and that its dimensions are
{d1,d2,d3,d4} then we need 4 positive integers a1,a2,a3,a4 to get
an element :
T[[a1,a2,a3,a4]]
and, of course ,we must have ai <= di for all i.
Transpose[ T, {2,1,3,2}]] is a tensor that given three positive
integers a1,a2,a3 will return the value
Transpose[ T, {2,1,3,2}] [[a1,a2,a3]] = T[[a2,a1,a3,a2]]
We only need three numbers because the repeated 2 uses the second
one twice. So the tensor rank is 3 and we must have a1 <= d2, a2
<= d1, a2 <= d4, a3 <= d3. In fact Transpose requires that d1 =
d4. The dimensions of the transposed tensor are {d2,d1,d4}
Similarly
Transpose[ T, {2,1,1,2}] [[a1,a2]] = T[[a2,a1,a1,a2]]
and the trnsposed tensor has rank 2 and dimensions {d2,d1}
Check
T = Array[SequenceForm,{2,4,4,2}];
Dimensions[T]
{2, 4, 4, 2}
Transpose[T,{2,1,3,2}][[1,2,3]]
2132
Dimensions[Transpose[T,{2,1,3,2}]]
{4, 2, 4}
Transpose[T,{2,1,1,2}][[1,2]]
2112
Dimensions[Transpose[T,{2,1,1,2}]]
{4, 2}
The GENERAL PATTERN (for elements) is
(**
Transpose[T_, p_List][[a__]] = T[[Sequence@@({a}[[p]])]]
**)
In the examples so far the length of p has been equal to the
tensor rank T (this is needed t o provide the right number of inputs
to T. But it would be inconvenient to always have to provide the
full sequence so for example, for a tensor T of tensor rank 5
Mathematica interprets
Transpose[ T, {2,1,1}] as Transpose[ T, {2,1,1,3,4}]
(appending successive integers from the first positive integer not
appearing in {2,1,1}).
So much for the elements.
But how do we construct the transposed tensor itself as an array?
Well that's the job of the Array function.
Notice that we don't need to allow for extending p -- this is
automatically taken care of
Trans[T_, p_List] :=
Block[{dims, trdims},
dims = Dimensions[T];
trdims =
dims[[Position[p,#,{1},1][[1,1]]]]&/@
Range[Length[Union[p]]];
Array[ T[[Sequence@@({##}[[p]])]]&, trdims ]
]
The conditions imposed by Transpose are (where the rank of T is r
and its dimensions are {d1,d2,...,dr }).
1. Union[ p] is Range[t] for some t <= r;
2. whenever pj = pk then dj = dk. (where p = {p1, p2, . . , pr})
The transposed tensor is of rank r minus the number of repeats in
p and, as for dimensions, if s occurs in at position t in p
then sth dimension is dt , otherwise the sth dimension can be
worked out by extending p as discusssed above.
Check
Trans[T,{2,1,1,2}] === Transpose[T,{2,1,1,2}]
Trans[T,{2,1,1}] === Transpose[T,{2,1,1}]
Trans[T,{1,2,2}] === Transpose[T,{1,2,2}]
True
True
True
Finding the diagonal
For a square matrix M, Transpose[M, {1,1}] is the diagonal of M
because.
Transpose[M, {1,1}][[a]] = M[[a,a]]
Thus
Transpose[{{a,A},{b,B}}, {1,1}]
{a, B}
But for non-square matrices
Transpose[{{a,A},{b,B},{c,C}}, {1,1}]
Transpose::perm: {1, 1} is not a permutation. (*message*)
Transpose[{{a, A}, {b, B}, {c, C}}, {1, 1}]
The message really means that the condition 2. is not met.
This can be allowed for by changing the definition of trdims in
the code for Trans
Trans2[T_, p_] :=
Block[{dims},
dims = Dimensions[T];
trdims =
Min[dims[[Flatten[Position[p,#,{1}]]]]]&/@
Range[Length[Union[p]]];
Array[ T[[Sequence@@({##}[[p]])]]&, trdims ]
]
Now we have
Trans2[{{a,A},{b,B},{c,C}}, {1,1}]
{a, B}
Trans2[Transpose[{{a,A},{b,B},{c,C}}], {1,1}]
{a, B}
It seems resaonable to call this the diagonal of these matrices
APPLICATION OF TRANSPOSE
Transpose[T,{1,1}] seems to be the quickest way to get the
diagonal of a square matrix.
The code for the determinant that I arrived at was
NewDet4[M_] :=
( Signature[M]
Plus@@Times@@
Append[ Transpose[#,{2,1,1}],(Signature/@#) ]&@
Permutations[ M ]
)
Timings
m = Table[ StringJoin["a",ToString[i],ToString[j]], {i,1,7},{j,1,7}];
NewDet4[m];//Timing
{6.58333 Second, Null}
Det[m];//Timing
{10.3667 Second, Null}
But be warned: I think that it was Dave Withoff who pointed out
that the result for Det was because of internal use of Expand and,
maybe,Together.
Block[{Expand},Det[m];//Timing]
{0.166667 Second, Null}
(Block[{Expand},Det[m]] - Det[m])//Expand
0
Allan Hayes
hay at haystack.demon.co.uk
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