He finds that he has drifted as far as the Odeon, one of the great world cafes, whose specialty is not listed anywhere—indeed has never been pinned down. Lenin, Trotsky, James Joyce, Dr. Einstein all sat out at these tables. Whatever it was they all had in common: whatever they’d come to this vantage to score … perhaps it had to do with the people somehow, with pedestrian mortality, restless crisscrossing of needs or desperations in one fateful piece of street… dialectics, matrices, archetypes all need to connect, once in a while, back to some of that proletarian blood, to body odors and senseless screaming across a table, to cheating and last hopes, or else all is dusty Dracularity, the West’s ancient curse… .
—  Gravity’s Rainbow, pg. 262-263, Thomas Pynchon
Quantum Physics

Conjugate transpose of matrices

Cont’d from “Dirac Notation: Introduction”, see “/tagged/matrices/

The transpose (denoted by a superscript ”T”) of a matrix 𝕄 with i rows and j columns, defined generally to be

is found by letting ijji (i.e. swapping the rows and columns) to get

This is often visualised by considering a ‘flip’ along the matrix’s trace (i.e. the diagonal elements where i = j), shown by the following animation from Wikipedia.

(Source: Wikimedia, Transpose (via Wikipedia)⇗)

Furthermore, the complex conjugate of this matrix is found by finding the complex conjugate of each of its elements individually. We denote the new matrix—the cojugate transpose—using a superscript “dagger”, †. Thus

where the complex conjugate of a complex number z = a + bi is found by letting i → − i to get z* = abi.


Il m'arrive de déborder de temps en temps et ça ne sera pas la première fois que je dérive du côté des LEGOS. Ici on peut voir le travail de Yoshi Akaï où la musique est composée en fonction de la construction de LEGO sur la matrice. Le son très synthétique prend une forme aussi abrupte que l'esthétique géométrique de la construction qui est aussi partition et plan de travail.

Je trouve l'idée très stimulante. On pourrait en parler longtemps et faire des ponts avec “Sonic Wire Sculptor”, une installation et un programme designé par Amit Pitaru dont j'espère bien vous parler dans un autre billet.