Matrix Decomposition by Discrete Fourier Transform

Given a 3-vector of complex numbers, (A,B,C), define its discrete Fourier transform as

where . That is, I’ll use lower case letters to denote the discrete Fourier transforms of UPPER case letters. The above leaves off a factor of but it will do.

Of interest today will be vectors (A,B,C) which happen to satisfy A+B+C = 0. These are eigenvectors of the Democratic D matrix

Democratic matrix with all entries D

that is, the matrix all of whose entries are equal to the complex number D. Of course their eigenvalues are zero. None of this is particularly interesting until we move from linearity to bilinearity and work with the discrete Fourier transforms of 3×3 matrices.

Define the Fourier transform of a 3×3 matrix U as where is the matrix:

Discrete Fourier transform matrix

where . With this definition, the discrete Fourier transform of the democratic matrix D, is:

Fourier transform of democratic matrix

This is a nice simplification.

Now let A+B+C=0 and compute some discrete Fourier transforms of four kinds of matrices, 1-circulant, 2-circulant, and two new types I will call “bra” and “ket” for obvious reasons. Untransformed matrices on the left, their transforms on the right, note that they fit together like the pieces of a jigsaw puzzle:

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An Immorality Tale

She was born with given name Johanna Maria Magdalena and a last name of either Behrend or Ritschel, my sources disagree. Her parents were unmarried, did she receive the last name of her father, Oskar Ritschel, or her mother, Auguste Behrend? In either case it was November 11, 1901. She was one of the most fascinating personalities of her time.

Youth
Her mother worked as a servant in Berlin and her father was an engineer who worked in various places around Europe. Soon after her birth, they married, but only for 3 years. Until she was 5, she stayed with her mother. Then she went to Belgium to visit her father who, after a delay of two years and insistent requests from the mother, finally told her that he had sent their child to be educated by the nuns at a convent (Catholic) boarding school in Brussels.

Her mother met and married a Jewish businessman, Richard Friedländer. When, the couple saw the conditions at the convent her mother decided to transfer her daughter to another convent, one that was less strict, in Vilvoorde, Belgium. Her parents moved to Schaerbeek, near Brussels (Belgium), and now she was able to come home to visit. With the marriage, she became Johanna Maria Magdalena Friedländer, and from the age of 7 she was raised in a household that observed both Catholic and Jewish customs.

In 1914, the world descended into the horror of the first world war. As German aliens living in Belgium, overnight the Friedländers became refugees. Eventually they made it to the German border, probably feeling fortunate that there was space available on a cattle car for them. As the modern world is one of passenger jets, the railroad was the transportation mode of the first half of the 20th century. Transport by livestock car is not a pleasant thing. Later, in the second world war, many thousands would be transported this way to the concentration camp at Buchenwald, where Richard Friedländer died. But let us return to her story.

Survivors at Buchenwald, April 16, 1945