Riddle: Transpose of Transpose

T

R

A

N

S

P

O

S

E

Also note that the transpose contains 9 letters and thus may be written as a square matrix:

TRA

NSP

OSE

which could be transposed to give

TNO

RSS

APE

or, reading across, left-to-right and top-to-bottom, 

"TNORSSAPE" 

which is the cipher-text you would transmit if you performed a scytale encryption on the clear-text TRANSPOSE

http://en.wikipedia.org/wiki/Scytale

We can see that the transpose is connected to seemingly unrelated fields, such as physics and cryptography, and must embody something fundamental indeed.

The Transpose of Products

Claim: For two matrices A,B, where the product of A and B is well defined, (AB)t = At Bt.

Proof: (AB)t = Bt At = At Bt.

It's a short proof, but the key point is that Bt and At commute because they are both matrices.

Using symmetry to look at the transpose

One way to define the transpose is to find the main diagonal of a matrix and transpose the matrix elements across it. Alternatively, we can know nothing about the process of finding the transpose and define the transpose from the definition of symmetric and antisymmetric matrices.

symmetric matrix = a matrix which equals its own transpose
antisymmetric matrix = a matrix which equals minus its own transpose

All square matrices can be broken down into the sum of a symmetric and an antisymmetric matrix (we need the fact that 2 transposes equals the identity). This means that if we have a group of matrices under addition which includes all symmetric and antisymmetric matrices, then we know that that group contains all square matrices.

I think we should spend more time investigating groups that are closed under transpose.

Comment on "A Recursive Method"

I don't think we can use the word "trivial" when taking the transpose of a matrix, especially a 2x2 matrix. Oftentimes, the word "trivial" is used synonymously with "too lazy to show." I feel that this blog should use more rigourous methods of proving properties of transposes. I liked the way you proved the transpose of a 1x1 matrix. If we start using the word "trivial" willy-nilly, I forsee a slippery slope.

I won't let you make a mockery of this blog, Evan.

A Recursive Method

Group the elements of the matrix into smaller matrices.  Transpose the location of the smaller matrices, and then transpose the elements of those matrices themselves.

This method has the advantage of being recursive, so we only need to know how to transpose certain shapes.  The base-cases for the recursion is simple:  single numbers and 2x2s.  The transpose of a single number is the number itself because you can treat it as a one-by-one matrix and thus the number itself is on the diagonal and does not move under transposition, and the transpose of a 2-by-2 is trivial.

Additionally, this implementing this method automatically implements the fact that if the matrix is block-diagonal, you can simply transpose the blocks.

You may think about taking this

If the previous posts in this blog interest you, you may consider taking this class offered by the UMBC philosophy department:

PHIL476 Modern Philosophy of Mathematical Operations; (3 credits) Grade Method: REG/P-F/AUD.

Prerequiste: PHIL100 (Introduction to Philosophy) and MATH763 (Algebraic Topology) OR MATH764 (Euclidean and Non-Euclidian Geometries)
A philosophical study of some of the current issues associated with mathematical operations and their interpretations in the physical world. The subject lies on the border between mathematics, physics, and philosophy. Topics include Banach and Hilbert interpretations, modern transpose theory, inertia group and factorial analysis. Credit will be granted for only one of the following: PHIL476 (Modern Philosophy of Mathematical Operations) or PHIL477 (Fourier Argument Theory)

Warm Up Exercises

1. Write a matrix down on a piece of paper.  By moving the paper only, can you find the transpose of the matrix?

2. Epistemology of Transposes:

a. If you take the transpose twice, should you have?

b. Prove that no matrix can have the property that taking the transpose twice doesn't give the matrix itself.  As a corollary, argue limitations on any all-powerful being, and discuss the implications of such a limitation on the struggle between good and evil, freedom and tyranny, Yankees and Red Sox, and other important religious ideas.

c. If someone takes a transpose of a symmetric matrix in the forest, did it make a noise?

3. Write a general 4x4 matrix in terms of the Dirac gamma matricies and their bilinears.  Prove, by the linearity of the transposition operation and direct transposition of those objects, that any 4x4 matrix may be transposed.

A physicist's doubt

The blog is still in its early stages, but we have seen some insightful methods of taking transposes. We also discussed an example, which involved matrices over the Complex Field (assuming that matrices with real entries were handled in an earlier course).

The discussion though has been quite mathematical till now. I, as an aspiring physicist, was wondering if there exists some physical interpretation to taking a transpose; or is it just interesting mathematics? Though I find the whole field of transpose math really fascinating, I still haven't come across any real physics in this area.

I hope this post would start a discussion and help us explore this fascinating technique further.

Hassam method : Noone can stop him from taking a transpose.

An example we went over in class

Ways to take the Transpose

1) Cohen's Method: Shut up and do it.

2) If the matrix is orthogonal, then find the inverse using

a) Gauss-Jordan elimination
b) Method of Cofactors

3) Iterate Cohen's methods 2n+1 times.

4) If the matrix is hermitian, take the complex conjugate.

more to come...