Logic Masters Deutschland e.V.

Puzzle link: Play on SudokuPad.

Rules: Normal Sudoku rules apply. Digits must not repeat within a cage.

Each cage labeled A_n represents a 2×2 matrix, with one of its eigenvectors contained in the cage labeled v_n and the associated eigenvalue in the cage labeled λ_n (n = 1,...,4). All eigenvectors and eigenvalues fulfill the eigenvalue equation A_nv_n = λ_nv_n, which is essentially written in matrix notation in the grid for each n, except that the equality sign and v_n on the right-hand side of the equation were omitted.

The multiplication of the matrix and its eigenvector, A_nv_n, is carried out in the usual way, i.e., the result in the top row is given by the sum of the top left digit of A_n multiplied with the top digit of v_n and the top right digit of A_n multiplied with the bottom digit of v_n. Analogously, the result of A_nv_n in the bottom row is given by the sum of the bottom left digit of A_n multiplied with the top digit of v_n and the bottom right digit of A_n multiplied with the bottom digit of v_n. For the multiplication λ_nv_n, the result in the top row is λ_n multiplied with the top digit of v_n, and in the bottom row it is λ_n multiplied with the bottom digit of v_n.

The cage labeled v_1a contains an alternative eigenvector of A₁ with associated eigenvalue λ_1a, either of which may be identical to or different from v₁ and λ₁, respectively.

The cage labeled λ₂+λ₃+λ₄ contains the sum of the three denoted eigenvalues in normal notation with the tens digit on the left and the ones digit on the right.

Example: The eigenvalue equation would be fulfilled for A = (6,2,3,1) with the elements listed as (top left, top right, bottom left, bottom right), v = (8,4) with the elements listed as (top, bottom), and λ = 7 because Av = (6×8+2×4,3×8+1×4) = (56,28), which equals λv = (7×8,7×4) = (56,28).

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Background: The properties of eigenvalues and eigenvectors are treated in introductory courses of linear algebra, but the puzzle is solvable without knowing that theory. Generally, neither eigenvalues nor eigenvectors need to consist of natural numbers, but the examples here were chosen such that this is the case.

The Schrödinger equation H ψ = E ψ is an example of an eigenvalue equation for which the Hamilton operator H is generally Hermitian, i.e., when written as a complex-valued matrix, it is equal to its conjugate transpose. As a consequence, the eigenvalues E, i.e., the energies of the quantum system, are real-valued. The eigenvectors ψ are the associated wave functions of the system at that energy. My prior puzzle on the Schrödinger equation visualized the fundamental relation of quantum mechanics. Although looking pretty, it does not contain the correct mathematical treatment. The current puzzle remedies this with four examples of eigenvalue equations in matrix notation.