Bohemian Matrices: Spectral and Structural Insights

• Bohemian matrices are families of matrices with entries drawn from a fixed finite population, exhibiting unique structural, spectral, and combinatorial behaviors.
• They are extensively studied in structured matrix theory, where constraints like symmetry, Toeplitz form, and Hessenberg structure lead to explicit recurrences and computational insights.
• Research on Bohemian matrices focuses on extremal spectral bounds, rapid characteristic height growth, and applications in exact computation and experimental mathematics.

Bohemian matrices are matrix families whose entries are drawn from a fixed finite population. In the original usage, the name is a mnemonic for BOunded HEight Matrix of Integers, but later work broadens the definition to matrices with entries in a fixed, usually discrete and hence bounded, subset of a field of characteristic zero; populations of integers, algebraic integers, and Gaussian integers all occur in the literature (Corless et al., 2022). The subject sits at the intersection of structured matrix theory, spectral geometry, combinatorics, symbolic computation, and experimental mathematics: finite populations make exhaustive or partially exhaustive computation possible, while structural constraints such as symmetry, upper Hessenberg form, Toeplitz form, or fixed subdiagonals generate nontrivial phenomena in characteristic polynomials, determinants, eigenvalue distributions, generalized inverses, and extremal problems (Chan et al., 2018).

1. Definition, terminology, and basic invariants

A Bohemian matrix family is specified by a population $P$ and, frequently, by additional structural constraints. In the standard formulation, every entry of every matrix in the family belongs to $P$, so at each fixed dimension the family is combinatorially finite. Several papers use populations such as

$$P=\{-1,0,+1\},\qquad P=\{0,1,i,-1,-i\},\qquad P=\{1,i\},$$

while symmetric endpoint-valued families are written as

$$S_m(\{a,b\})=\{\,m\times m \text{ symmetric matrices with entries only from }\{a,b\}\,\}$$

and compared with the larger interval class

$$S_m([a,b])=\{\,m\times m \text{ symmetric matrices with entries in the real interval }[a,b]\,\}$$

(Chan et al., 2019, Calkin et al., 30 Sep 2025).

The notion of height is central. For a matrix $A$, the height is the infinity norm of the vector obtained by reshaping $A$, and for Bohemian populations such as $\{-1,0,1\}$ this matrix height is constant: $\mathrm{height}(A)=1.$ The associated characteristic height is the maximum absolute value among the coefficients of the characteristic polynomial. One of the recurrent themes of the subject is the contrast between bounded entry height and rapidly growing characteristic height (Chan et al., 2019).

Bohemian matrices are also linked to the classical notion of height for polynomials. Because every polynomial in the monomial basis can be embedded as a Frobenius companion matrix, root questions for bounded-height polynomials naturally translate into eigenvalue questions for Bohemian matrices. At the same time, several Bohemian phenomena do not reduce to companion-matrix questions, especially when the family is constrained by symmetry, Toeplitz structure, or Hessenberg sparsity (Corless et al., 2022).

2. Structured families and representative constructions

Much of the theory concentrates on structured subclasses. A major example is the upper Hessenberg family, characterized by zeros below the first subdiagonal. In one standard form,

$$\mathbf{H}_n = \begin{bmatrix} h_{1,1} & h_{1,2} & \cdots & h_{1,n}\ s & h_{2,2} & \cdots & h_{2,n}\ 0 & s & \ddots & \vdots\ \vdots & \ddots & \ddots & h_{n-1,n}\ 0 & \cdots & 0 & s & h_{n,n} \end{bmatrix},$$

the upper-triangular entries are drawn from a population such as $P$0, while the subdiagonal entries are fixed roots of unity, typically $P$1 or $P$2. A further specialization is the upper Hessenberg Toeplitz family

$P$3

which retains finite-population combinatorics while reducing the degrees of freedom to one parameter per diagonal (Chan et al., 2018).

Another representative construction is the skew-symmetric pentadiagonal family used in computational-discovery settings: $P$4 If $P$5, each of the seven free parameters has two choices, so the $P$6 family contains $P$7 matrices (Calkin et al., 2021).

The theory also includes arithmetic constructions. A recent example studies $P$8 integer Bohemian matrices

$P$9

with the property that all 24 permutations of the entries $$P=\{-1,0,+1\},\qquad P=\{0,1,i,-1,-i\},\qquad P=\{1,i\},$$0 yield matrices with integer eigenvalues. Under the ansatz

$$P=\{-1,0,+1\},\qquad P=\{0,1,i,-1,-i\},\qquad P=\{1,i\},$$1

the six discriminant conditions collapse to a Pythagorean-triple mechanism, and the triple $$P=\{-1,0,+1\},\qquad P=\{0,1,i,-1,-i\},\qquad P=\{1,i\},$$2 produces the coefficient set

$$P=\{-1,0,+1\},\qquad P=\{0,1,i,-1,-i\},\qquad P=\{1,i\},$$3

for which every permutation has integer spectrum (Hall, 3 Jan 2026).

A different recent direction concerns structured classes over $$P=\{-1,0,+1\},\qquad P=\{0,1,i,-1,-i\},\qquad P=\{1,i\},$$4, including rank-one Bohemian matrices and higher-rank Classes I, II, and III. These are organized by decompositions into rank-one Bohemian pieces and by disjoint-support or block-orthogonality conditions, yielding canonical forms for generalized-inverse problems (Chowdhry et al., 23 Aug 2025).

3. Characteristic polynomials, recurrences, and coefficient growth

Upper Hessenberg structure yields explicit recurrences for characteristic polynomials. If

$$P=\{-1,0,+1\},\qquad P=\{0,1,i,-1,-i\},\qquad P=\{1,i\},$$5

then

$$P=\{-1,0,+1\},\qquad P=\{0,1,i,-1,-i\},\qquad P=\{1,i\},$$6

Writing

$$P=\{-1,0,+1\},\qquad P=\{0,1,i,-1,-i\},\qquad P=\{1,i\},$$7

one obtains coefficient recurrences as well. In the Toeplitz specialization these simplify to

$$P=\{-1,0,+1\},\qquad P=\{0,1,i,-1,-i\},\qquad P=\{1,i\},$$8

together with the induced recurrence for the coefficients $$P=\{-1,0,+1\},\qquad P=\{0,1,i,-1,-i\},\qquad P=\{1,i\},$$9 (Chan et al., 2018, Chan et al., 2018).

These recurrences encode substantial combinatorics. For the constant terms, small cases expand as

$$S_m(\{a,b\})=\{\,m\times m \text{ symmetric matrices with entries only from }\{a,b\}\,\}$$0

$$S_m(\{a,b\})=\{\,m\times m \text{ symmetric matrices with entries only from }\{a,b\}\,\}$$1

and the coefficients match the combinatorics of integer compositions. In pedagogical settings, these expansions are used as discovery problems: after computing $$S_m(\{a,b\})=\{\,m\times m \text{ symmetric matrices with entries only from }\{a,b\}\,\}$$2 and $$S_m(\{a,b\})=\{\,m\times m \text{ symmetric matrices with entries only from }\{a,b\}\,\}$$3 determinants, one is asked to infer the pattern for larger sizes (Calkin et al., 2021, Chan et al., 2018).

A central invariant is the characteristic height. For upper Hessenberg Toeplitz matrices with $$S_m(\{a,b\})=\{\,m\times m \text{ symmetric matrices with entries only from }\{a,b\}\,\}$$4, maximal characteristic height occurs when

$$S_m(\{a,b\})=\{\,m\times m \text{ symmetric matrices with entries only from }\{a,b\}\,\}$$5

and for $$S_m(\{a,b\})=\{\,m\times m \text{ symmetric matrices with entries only from }\{a,b\}\,\}$$6 it is also attained by the alternating pattern

$$S_m(\{a,b\})=\{\,m\times m \text{ symmetric matrices with entries only from }\{a,b\}\,\}$$7

In the extremal all-$$S_m(\{a,b\})=\{\,m\times m \text{ symmetric matrices with entries only from }\{a,b\}\,\}$$8 case, all coefficients of the characteristic polynomial are positive, and the constant term satisfies

$$S_m(\{a,b\})=\{\,m\times m \text{ symmetric matrices with entries only from }\{a,b\}\,\}$$9

so the maximal characteristic height is at least $$S_m([a,b])=\{\,m\times m \text{ symmetric matrices with entries in the real interval }[a,b]\,\}$$0, hence exponential in the order (Chan et al., 2018).

The upper Hessenberg framework is also rigid in algebraic ways. Every matrix in the upper Hessenberg Bohemian families with nonzero subdiagonal is non-derogatory, so the characteristic polynomial and minimal polynomial coincide up to a sign. In the Toeplitz upper Hessenberg family, each matrix has a unique characteristic polynomial; the number of distinct characteristic polynomials therefore equals the number of matrices in the family (Chan et al., 2018, Chan et al., 2019).

4. Spectral geometry and extremal spectral questions

A distinctive feature of Bohemian matrix theory is the geometry of eigenvalue sets. For dense unstructured random matrices with entries from a discrete population such as $$S_m([a,b])=\{\,m\times m \text{ symmetric matrices with entries in the real interval }[a,b]\,\}$$1, Tao and Vu showed that the eigenvalues of $$S_m([a,b])=\{\,m\times m \text{ symmetric matrices with entries in the real interval }[a,b]\,\}$$2 matrices become asymptotically uniformly distributed on a disk of radius $$S_m([a,b])=\{\,m\times m \text{ symmetric matrices with entries in the real interval }[a,b]\,\}$$3. Structured Bohemian families behave very differently: the literature reports strips, squares, diamonds, irregular hexagonal regions, and Toeplitz spectral curves instead of disk-like supports (Corless et al., 2022).

Several exact spectral bounds are known. If $$S_m([a,b])=\{\,m\times m \text{ symmetric matrices with entries in the real interval }[a,b]\,\}$$4 is complex symmetric of dimension $$S_m([a,b])=\{\,m\times m \text{ symmetric matrices with entries in the real interval }[a,b]\,\}$$5 with entries drawn from $$S_m([a,b])=\{\,m\times m \text{ symmetric matrices with entries in the real interval }[a,b]\,\}$$6, then every eigenvalue $$S_m([a,b])=\{\,m\times m \text{ symmetric matrices with entries in the real interval }[a,b]\,\}$$7 satisfies

$$S_m([a,b])=\{\,m\times m \text{ symmetric matrices with entries in the real interval }[a,b]\,\}$$8

For square skew-symmetric matrices with population $$S_m([a,b])=\{\,m\times m \text{ symmetric matrices with entries in the real interval }[a,b]\,\}$$9,

$A$0

For skew-symmetric tridiagonal matrices with population $A$1 and $A$2, the eigenvalues lie in the diamond

$A$3

For unit upper Hessenberg zero-diagonal matrices with $A$4, every eigenvalue satisfies the dimension-independent bound

$A$5

(Corless et al., 2022).

Experiments reveal additional phenomena that remain unexplained. One widely cited figure plots the eigenvalue density of all

$A$6

ten-by-ten Bohemian skew-pentadiagonal matrices with population $A$7 in the rectangle

$A$8

Hotter colors indicate higher density, and the authors explicitly state that at the time of writing they had “no explanation whatever for the distribution pattern visible here” (Calkin et al., 2021). In the Toeplitz upper Hessenberg setting, the eigenvalues of all matrices of fixed size likewise appear to form an irregular hexagonal region, again with apparent symmetries and density flecks for which the authors report no explanation (Chan et al., 2018).

The spread problem provides an extremal spectral counterpart. For a real symmetric matrix $A$9,

$A$0

Fallat and Xing conjectured that the maximum spread over $A$1 is attained by a rank-$A$2 matrix in $A$3. Biborski reduced the search to the endpoint-valued Bohemian family, and recent work treats the problem computationally through the resultant

$A$4

whose roots are precisely the eigenvalue differences. The conjecture is proved for $A$5 with $A$6, for $A$7 with $A$8 and $A$9, and for $\{-1,0,1\}$0 (Calkin et al., 30 Sep 2025).

5. Determinants, normality, stability, and Bohemian inverses

Determinant extremals form another major branch of the subject. For

$\{-1,0,1\}$1

the family of upper Hessenberg matrices with fixed subdiagonal entries $\{-1,0,1\}$2 and upper-triangular entries in $\{-1,0,1\}$3, the determinant-maximization problem is solved in several regimes. In particular, for $\{-1,0,1\}$4, the maximum absolute determinant $\{-1,0,1\}$5 over matrices with upper-triangular entries in $\{-1,0,1\}$6 satisfies

$\{-1,0,1\}$7

In the discrete Bohemian case $\{-1,0,1\}$8 with $\{-1,0,1\}$9, this becomes

$\mathrm{height}(A)=1.$0

proving a conjecture of Fasi and Negri Porzio. The extremal matrix is the alternating pattern $\mathrm{height}(A)=1.$1, with $\mathrm{height}(A)=1.$2 when $\mathrm{height}(A)=1.$3 and $\mathrm{height}(A)=1.$4 is even, and $\mathrm{height}(A)=1.$5 when $\mathrm{height}(A)=1.$6 and $\mathrm{height}(A)=1.$7 is odd (Keating et al., 2020).

The same structured families exhibit strong restrictions on normality and stability. In zero-diagonal upper Hessenberg families with population $\mathrm{height}(A)=1.$8, $\mathrm{height}(A)=1.$9, the only normal matrices for $$\mathbf{H}_n = \begin{bmatrix} h_{1,1} & h_{1,2} & \cdots & h_{1,n}\ s & h_{2,2} & \cdots & h_{2,n}\ 0 & s & \ddots & \vdots\ \vdots & \ddots & \ddots & h_{n-1,n}\ 0 & \cdots & 0 & s & h_{n,n} \end{bmatrix},$$0 are symmetric, $$\mathbf{H}_n = \begin{bmatrix} h_{1,1} & h_{1,2} & \cdots & h_{1,n}\ s & h_{2,2} & \cdots & h_{2,n}\ 0 & s & \ddots & \vdots\ \vdots & \ddots & \ddots & h_{n-1,n}\ 0 & \cdots & 0 & s & h_{n,n} \end{bmatrix},$$1-skew symmetric, or $$\mathbf{H}_n = \begin{bmatrix} h_{1,1} & h_{1,2} & \cdots & h_{1,n}\ s & h_{2,2} & \cdots & h_{2,n}\ 0 & s & \ddots & \vdots\ \vdots & \ddots & \ddots & h_{n-1,n}\ 0 & \cdots & 0 & s & h_{n,n} \end{bmatrix},$$2-skew circulant. For zero-diagonal upper Hessenberg Bohemian families, Type I stability is impossible because the trace is zero, while for integer Bohemian matrices Type II stability is equivalent to nilpotence (Chan et al., 2018).

Recent work has extended the subject to generalized inverses. For $$\mathbf{H}_n = \begin{bmatrix} h_{1,1} & h_{1,2} & \cdots & h_{1,n}\ s & h_{2,2} & \cdots & h_{2,n}\ 0 & s & \ddots & \vdots\ \vdots & \ddots & \ddots & h_{n-1,n}\ 0 & \cdots & 0 & s & h_{n,n} \end{bmatrix},$$3, an inner inverse satisfies

$$\mathbf{H}_n = \begin{bmatrix} h_{1,1} & h_{1,2} & \cdots & h_{1,n}\ s & h_{2,2} & \cdots & h_{2,n}\ 0 & s & \ddots & \vdots\ \vdots & \ddots & \ddots & h_{n-1,n}\ 0 & \cdots & 0 & s & h_{n,n} \end{bmatrix},$$4

and an outer inverse satisfies

$$\mathbf{H}_n = \begin{bmatrix} h_{1,1} & h_{1,2} & \cdots & h_{1,n}\ s & h_{2,2} & \cdots & h_{2,n}\ 0 & s & \ddots & \vdots\ \vdots & \ddots & \ddots & h_{n-1,n}\ 0 & \cdots & 0 & s & h_{n,n} \end{bmatrix},$$5

Restricting $$\mathbf{H}_n = \begin{bmatrix} h_{1,1} & h_{1,2} & \cdots & h_{1,n}\ s & h_{2,2} & \cdots & h_{2,n}\ 0 & s & \ddots & \vdots\ \vdots & \ddots & \ddots & h_{n-1,n}\ 0 & \cdots & 0 & s & h_{n,n} \end{bmatrix},$$6 to the same population $$\mathbf{H}_n = \begin{bmatrix} h_{1,1} & h_{1,2} & \cdots & h_{1,n}\ s & h_{2,2} & \cdots & h_{2,n}\ 0 & s & \ddots & \vdots\ \vdots & \ddots & \ddots & h_{n-1,n}\ 0 & \cdots & 0 & s & h_{n,n} \end{bmatrix},$$7 defines inner and outer Bohemian inverses. For rank-one matrices $$\mathbf{H}_n = \begin{bmatrix} h_{1,1} & h_{1,2} & \cdots & h_{1,n}\ s & h_{2,2} & \cdots & h_{2,n}\ 0 & s & \ddots & \vdots\ \vdots & \ddots & \ddots & h_{n-1,n}\ 0 & \cdots & 0 & s & h_{n,n} \end{bmatrix},$$8, the nonzero outer inverses are completely characterized by

$$\mathbf{H}_n = \begin{bmatrix} h_{1,1} & h_{1,2} & \cdots & h_{1,n}\ s & h_{2,2} & \cdots & h_{2,n}\ 0 & s & \ddots & \vdots\ \vdots & \ddots & \ddots & h_{n-1,n}\ 0 & \cdots & 0 & s & h_{n,n} \end{bmatrix},$$9

Beyond rank one, complete characterizations are obtained for the inner inverses of Class III matrices and full-row rank Class II matrices, and for the full set of outer inverses of rank-two full-row rank Class III matrices (Chowdhry et al., 23 Aug 2025).

6. Computational discovery, exact computation, and open questions

Bohemian matrices have become a model setting for computational discovery. In one explicit teaching-and-research framework, the workflow is: define a structured family with entries from a finite population; compute determinants, eigenvalues, characteristic polynomials, and singularity counts; search for patterns; visualize the data; and pose new questions. Typical questions include how many matrices belong to the family, how many distinct characteristic polynomials occur, how many distinct eigenvalues occur, how many matrices are singular, and how quickly one can generate all $P$00 matrices when $P$01 (Calkin et al., 2021).

This use is not merely pedagogical. The literature repeatedly emphasizes that the area is new and largely unexplored, that many natural questions remain open, and that even small structured families can yield genuinely new mathematics. The Bohemian matrices component of the computational-discovery program is described as especially effective because there are “so many interesting pictures already,” because “there is so much unknown,” and because some of the displayed images were created by students and had “never been seen by anyone before” (Calkin et al., 2021).

A parallel methodological theme is the tension between numerical and exact computation. Numerical eigenvalue plots are indispensable for discovering structure, but exact computation is often preferable when multiple eigenvalues or near-multiplicities are present. Accordingly, Bohemian matrix research makes extensive use of characteristic-polynomial recurrences, exact algebraic computation, resultants, Sturm sequences, Routh–Hurwitz tests, Gershgorin arguments, Schmidt–Spitzer theory, graph isomorphism and canonical labeling, and exact arithmetic in Maple (Corless et al., 2022, Calkin et al., 30 Sep 2025).

Several open directions remain explicit in the literature. Spectral density patterns for skew-pentadiagonal and Toeplitz upper Hessenberg families are still only partially understood. Extremal spread is settled only in specific cases. Characteristic-height asymptotics are known to be exponential, but the full asymptotic picture remains incomplete. A plausible implication is that Bohemian matrices will continue to function as a tractable research laboratory in which combinatorial finiteness coexists with unresolved questions in spectral theory, structured linear algebra, and exact computation.