Matrices with integer eigenvalues for all permutations of coefficients (thanks to Pythagoras!)

1. Problem Statement

The paper addresses the problem of generating $2 \times 2$ matrices with integer coefficients ($a, b, c, d$) such that every possible permutation of these four coefficients results in a matrix with integer eigenvalues.

While methods exist to generate matrices with integer eigenvalues for a specific arrangement, the challenge lies in finding sets of integers $\{a, b, c, d\}$ where the eigenvalue property holds invariant under all $24$ possible permutations of the matrix entries. This is motivated by educational contexts (e.g., classroom exercises) where a teacher needs to distribute different matrices to students using the same set of numbers, ensuring all yield integer solutions.

2. Mathematical Methodology

The Eigenvalue Condition

For a $2 \times 2$ matrix $M = \begin{pmatrix} a & b \\ c & d \end{pmatrix}$, the eigenvalues $\lambda$ are roots of the characteristic equation:
$$\lambda^2 - (a+d)\lambda + (ad-bc) = 0$$
The eigenvalues are integers if and only if the discriminant is a perfect square, say $u^2$. This leads to the fundamental condition:
$$u^2 = (a+d)^2 - 4(ad-bc) = (a-d)^2 + 4bc$$
For all permutations of $\{a, b, c, d\}$ to yield integer eigenvalues, six specific Diophantine equations must be satisfied simultaneously (corresponding to the six distinct ways to pair the coefficients for the discriminant):

1. $u^2 = (a-d)^2 + 4bc$
2. $v^2 = (b-c)^2 + 4ad$
3. $w^2 = (a-b)^2 + 4cd$
4. $x^2 = (c-d)^2 + 4ab$
5. $y^2 = (a-c)^2 + 4bd$
6. $z^2 = (b-d)^2 + 4ac$

The Ansatz and Reduction

The author introduces a simplifying assumption (ansatz) to solve this system:
$$a + d = b + c = t$$
where $t$ is an integer.

• Under this assumption, conditions (3) through (6) are automatically satisfied.
• Conditions (1) and (2) simplify to:
  $$u^2 = t^2 + 4(a-b)(b-d)$$
  $$v^2 = t^2 - 4(a-b)(b-d)$$
• By defining $k = b-d$ and $l = a-b$, the problem reduces to finding integers $r, s, t$ such that:
  $$r^2 + s^2 = t^2$$
  This reveals that the solution space is directly linked to Pythagorean triples $(r, s, t)$.

Derivation of Coefficients

Using the relations derived from the ansatz, the four matrix coefficients are expressed in terms of the Pythagorean triple $(r, s, t)$ and factors $k, l$ (where $kl = rs/2$):
$$\{a, b, c, d\} = \left\{ \frac{t \pm k \pm l}{2} \right\}$$
To ensure $a, b, c, d$ are integers, the author proposes a canonical solution by choosing $k=r$ and $l=s/2$ (leveraging the parity properties of Pythagorean triples where one leg is divisible by 4). This yields the set:
$$\{a, b, c, d\} = \left\{ \frac{t \pm r \pm \frac{s}{2}}{2} \right\}$$

3. Key Contributions and Results

A. The Canonical Solution

The paper provides a direct formula to generate integer sets from any Pythagorean triple $(r, s, t)$.

• Example 1: Triple $(3, 4, 5)$ generates $\{5, 3, 2, 0\}$.
• Example 2: Triple $(5, 12, 13)$ generates $\{12, 6, 7, 1\}$.
  Any permutation of these numbers into a $2 \times 2$ matrix yields integer eigenvalues.

B. Alternative Parametric Forms

The author derives a more user-friendly parametric form using coprime integers $m$ and $n$ (with $n$ odd):
$$\{a, b, c, d\} = \{4m^2 \pm mn, \ n^2 \pm mn\}$$
This form recovers the canonical examples and allows for easy generation of strictly positive integer coefficients (provided $4|m| > |n| > |m|$).

C. Infinite Families of Solutions

The paper demonstrates that a single Pythagorean triple can generate a countable infinity of non-trivial solutions.

• By choosing non-integer rational factors for $k$ and $l$ in the general formula and scaling by a common factor, new integer sets are produced.
• Example: From triple $(3, 4, 5)$, choosing specific rational $k, l$ yields sets like $\{21, 15, 5, -1\}$ and $\{65, 49, 11, -5\}$.

D. Explicit Eigenvalue Formulas

The paper explicitly calculates the integer eigenvalues for the various permutations of the generated matrices. For the canonical solution, the eigenvalues take simple forms such as:
$$\left\{ \frac{t \pm (r+s)}{2} \right\}, \quad \left\{ \frac{t \pm (r-s)}{2} \right\}, \quad \{t, k\}, \quad \{t, -k\}, \dots$$
Crucially, the eigenvalues for the first two matrix permutations are independent of the specific choice of $k$ and $l$.

4. Significance

• Pedagogical Utility: The method provides a robust tool for educators to generate large sets of distinct problems (matrices) that share the same coefficients but yield integer solutions, facilitating quick classroom exercises without complex calculation.
• Mathematical Insight: It establishes a novel and elegant connection between Pythagorean triples and the spectral properties of matrices under permutation. It shows that the symmetry required for all permutations to have integer eigenvalues is deeply rooted in the geometry of right-angled triangles.
• Efficiency: Unlike previous methods that generate one matrix per parameter set, this method generates a "family" of 24 matrices from a single set of parameters.
• Generalization: The paper opens the door for further research, noting that while the $2 \times 2$ case is solved via this ansatz, the existence of non-trivial solutions for $3 \times 3$ matrices remains an open question.

Conclusion

Michael J.W. Hall successfully derives a constructive method to generate $2 \times 2$ matrices with integer coefficients and integer eigenvalues for all $24$ permutations of their entries. The core innovation is the reduction of the problem to Pythagorean triples, offering both a canonical solution and an infinite family of variations, thereby solving a specific combinatorial matrix problem with significant educational and theoretical implications.