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This post is mainly a reminder that we should continue to post awesome! jokes on this blog:

And now a terrible joke...


Q: What is the value of the contour integral around Western Europe?







A: Zero, because all the Poles are in Eastern Europe.

Symmetry of Lorentz Boost

The Lorentz transformation matrix is symmetric under transpose!

Saving Lives with the Transpose

While doing my usual transpose research, I stumbled upon this definition from the American Heritage Medical Dictionary:

transpose (redirected from "Transpose of a Matrix")

v. - to transfer one tissue, organ, or part to the place of another.


A natural extension of this is to write a matrix representation of the body, where each element in the matrix represents a location in the body. Then fill in the matrix with body parts, take the transpose, and that should tell you how to take the transpose of the human body. Try it out and let me know how it goes.



(Taken from: http://74.125.45.132/search?q=cache:ZLoWQTsWrK4J:medical-dictionary.thefreedictionary.com/Transpose%2Bof%2Ba%2Bmatrix+how+to+take+the+transpose+of+a+matrix&hl=en&ct=clnk&cd=10&gl=us&client=firefox-a)

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.