Is Matrix Multiplication Ugly?

Overall view on “ugliness”

• Many consider matrix multiplication aesthetically pleasing or at least neutral; calling it ugly is seen as mostly a matter of personal taste.
• Several argue that noncommutativity alone is not a valid aesthetic objection; by that standard, subtraction and division would also be “ugly.”
• Some agree that, subjectively, matrix multiplication feels clunky or unintuitive, often tied to their own limited comfort with linear algebra.

Noncommutativity & representation

• Multiple comments note that the noncommutativity issue is about transformations, not matrices specifically: rotations, function composition, and “socks then shoes vs shoes then socks” all naturally fail to commute.
• Matrices are described as a sometimes ugly representation of beautiful objects (linear transformations), akin to a crude “map of elegant territory.”
• Others say matrices themselves are ugly because they bake in an arbitrary choice of basis, though linear transformations are “beautiful.”

Matrix multiplication in AI and applications

• Some find it strikingly beautiful that chaining matmuls over huge tensors yields systems that can converse, reason, and generate media.
• Others see AI’s endless matmuls as brute-force mixing of every input component with every other, inefficient relative to how brains likely work.
• There is concern about scaling: burning gigawatts on dense matmuls may be unsustainable; transformers might be an elaborate dead end if more efficient architectures emerge.

Algorithms, structure, and efficiency

• Discussion references faster-than–O(n³) algorithms (Strassen and successors) and frustration that we still lack a clean answer on the true complexity exponent.
• Structured matrices (low-rank, block-diagonal, Monarch-style factorizations, FFT as sparse factorization) are presented as more “beautiful” and far more parameter/FLOP efficient, and sometimes used in practice (LoRA, attention variants).
• Some complain that matmul libraries are “ugly” in interface or trade-offs, even if GEMM is hardware-friendly.

Pedagogy, terminology, and intuition

• The phrase “send (x, y) to (−x, y)” confused some readers; rephrasing it as “change the sign of x” was seen as clearer.
• Several recommend thinking of matrix multiplication as composition of linear maps, which makes its form and noncommutativity feel natural.
• Overloading the word “multiplication” (matrices, dot, cross, Hadamard) is blamed for confusion; some suggest a distinct term might have helped.

Beauty vs usefulness

• One camp dismisses “elegance” as irrelevant: what matters is solving real problems, not mathematical aesthetics.
• Another insists that for working mathematicians, notions like simplicity, parsimony, and “good explanations” are central—and that much of modern math is pursued primarily for its beauty.