A = UΣVᵀ
Every real matrix (square or rectangular, full-rank or singular) factors into a pair of orthogonal matrices flanking a rectangular diagonal of nonnegative singular values. The left and right singular vectors rotate the domain and codomain so that the linear map becomes a pure stretch along aligned axes; the singular values themselves are those stretches.
When the values decay rapidly the map is essentially low-dimensional, and the partial sum formed by the first k terms is the nearest rank-k matrix in every unitarily invariant norm. Beltrami obtained the square case in 1873, Jordan independently a year later.
One factorization therefore exposes geometry, numerical rank, and the most economical compressions of the data the matrix represents.
In the 1940s, Subrahmanyan Chandrasekhar was committed to his teaching role at the University of Chicago, despite being based at the Yerkes Observatory. Each week, he traveled 80 miles to teach a special course attended by only two students.
The students were Tsung-Dao Lee and Chen-Ning Yang. They proved their mentor's faith was well-placed when they both won the Nobel Prize in Physics in 1957, years before Chandrasekhar received the same honor in 1983. Remarkably, this course went down in history as the only one where every attendee received a Nobel Prize, underscoring the extraordinary impact of Chandrasekhar's dedication and teaching.
📷 AIP Emilio Segrè Visual Archives, Physics Today Collection
A tesseract is the four-dimensional analog of a cube, also known as a hypercube in 4-dimensional space. Just as a cube has six square faces, a tesseract has eight cubic "faces" in 4-dimensional space.
BREAKING 🚨: The Sun unleashed an intense X-class solar flare that's triggering radio blackouts across Australia
A CME was also released, but early analyses of satellite coronagraph imagery suggest it's not Earth-directed