Archetypes: Reference Catalogue of Worked Systems

Overview

Theory in linear algebra is stated in generality, but understanding of it is built on particular cases. A catalogue of canonical examples serves that purpose only if it is designed: the entries must span the range of qualitatively different behaviours, they must be worked out to a common and complete list of properties, and enough of them must be related to one another that comparison is informative. A pile of random matrices satisfies none of these conditions.

This catalogue has twenty-three entries labelled A through W. Entries A to J are systems of linear equations, ranging from three equations in three unknowns to six equations in nine unknowns, and covering consistent systems with a unique solution, consistent systems with infinitely many, and inconsistent systems. Entries K and L are square matrices presented for their own sake, one nonsingular and one singular. Entries M to W are linear transformations between spaces of column vectors, matrices and polynomials, chosen so that every combination of injective and surjective that dimension permits actually occurs.

Each entry is worked through the same property list: the system and sample solutions where applicable, the augmented matrix and its reduced form, the analysis symbols $r$ , $D$ and $F$ , the vector form of the solution set, the associated homogeneous system, the coefficient matrix and its reduced form, singularity, the four fundamental subspaces with explicit bases, the inverse where it exists, rank and nullity, the determinant, the eigenvalues with eigenspace bases, the two multiplicities, and diagonalisability. Applying one uniform template to every entry is what makes cross-entry comparison meaningful.

The most valuable feature is the deliberate pairing. Entries D and E share a coefficient matrix and differ only in the vector of constants; one is consistent and one is not, which demonstrates that consistency is a property of the pair rather than of the matrix. Entries G and H are paired the same way at a different shape, five equations in two unknowns. Entries A and B are both three-by-three systems, one with a singular coefficient matrix and one nonsingular. These controlled comparisons do work that no amount of unrelated examples can do.

Definition

Archetype

ARCH

An archetype in this catalogue is a fully analysed instance — a system of linear equations, a matrix, or a linear transformation — worked through a fixed and complete list of structural properties. Entries are labelled by single letters, A through W, and referred to by label throughout the library.

The entries are deliberately larger and less tidy than the illustrative examples embedded in a discussion, so that patterns which happen to hold at size two or three but fail in general are not reinforced.

Paired Entries

ARCH-P

Two entries form a pair when they agree in every respect but one. In this catalogue, D and E share a coefficient matrix and differ only in the vector of constants; G and H do the same at a different size. A pair isolates one structural variable and shows precisely which conclusions depend on it.

This is the mathematical form of a controlled comparison. A single example can illustrate a phenomenon; only a pair can demonstrate what causes it.

Property List

ARCH-PL

The fixed schedule of computed quantities recorded for every entry: reduced row-echelon forms of both the augmented and the coefficient matrix; the analysis symbols $r$ , $D$ and $F$ ; the vector form of the solution set; singularity; bases for $\mathcal{N}(A)$ , $\mathcal{C}(A)$ , $\mathcal{R}(A)$ and $\mathcal{L}(A)$ ; the inverse if it exists; $\operatorname{rank}(A)$ and $\operatorname{nullity}(A)$ ; $\det(A)$ ; the eigenvalues with eigenspace bases; $\alpha_A(\lambda)$ and $\gamma_A(\lambda)$ ; and diagonalisability.

Entries that are not square omit the properties that require squareness — determinant, inverse, eigenvalues — rather than substituting a placeholder. Which properties are absent is itself information about the entry.

Concepts

Coverage of the system possibilities

A system of linear equations admits exactly three outcomes: no solution, one solution, or infinitely many. The system entries realise all three at several shapes. Entry B is three equations in three unknowns with a nonsingular coefficient matrix and a unique solution. Entry A is the same shape with a singular coefficient matrix, rank $2$ and nullity $1$ , so its solution set is a one-parameter family. Entry E is inconsistent, and entry H is inconsistent while being overdetermined at five equations in two unknowns. Entries I and J push the shape the other way, at four equations in seven unknowns and six in nine, with nullities of four and five respectively.

The D and E pair: consistency belongs to the pair, not the matrix

Entries D and E have identical coefficient matrices, three by four, of rank $2$ . Their augmented matrices reduce differently: D's reduced form has two non-zero rows with no leading one in the final column, so the system is consistent with $4 - 2 = 2$ degrees of freedom; E's has three non-zero rows with a leading one in column $5$ , encoding the contradiction $0 = 1$ . Every property that depends only on the coefficient matrix — rank, nullity, the four subspaces, the null space basis — is identical for the two entries. Only the solvability differs. This is the cleanest available demonstration that reducing $A$ alone discards exactly the information needed to detect inconsistency.

The G and H pair at an overdetermined shape

Entries G and H are five equations in two unknowns sharing a coefficient matrix of rank $2$ and nullity $0$ . Because the nullity is zero, consistency immediately forces uniqueness: G is consistent and has the single solution $(6, -2)$ , with three of its five equations redundant. H, with a different constant vector, is inconsistent, and its augmented matrix reduces to a leading one in the third column. The pair shows that having more equations than unknowns neither guarantees inconsistency nor forbids it — a point that a single overdetermined example would leave ambiguous.

Matrix entries: multiplicity structure and diagonalisability

Entries K and L are both five by five and both diagonalisable, but for different reasons. K is nonsingular with determinant $16$ and three distinct eigenvalues, two of them of algebraic multiplicity two, each with matching geometric multiplicity. L is singular with rank $3$ and nullity $2$ , and has just two distinct eigenvalues of algebraic multiplicities three and two, again matching geometrically. Contrast entry A, whose coefficient matrix has eigenvalue $0$ with algebraic multiplicity two but geometric multiplicity one: a defective eigenvalue, and therefore not diagonalisable. Three entries, three multiplicity patterns, one conclusion — diagonalisability is decided by the gap between the two multiplicities and by nothing else.

Transformation entries: what dimension forbids and what it merely permits

For a linear transformation, the relative dimensions of domain and codomain rule out one property outright and leave the other open. When the domain is larger, injectivity is impossible; when the codomain is larger, surjectivity is impossible. The transformation entries are arranged to occupy every remaining cell. Entry M maps a five-dimensional space to a three-dimensional one with rank $2$ and is therefore neither; entry N has the same shape but rank $3$ and is surjective. Entry O maps three dimensions into five with rank $2$ and is neither; entry P has the same shape with rank $3$ and is injective. Entry R is a transformation of a five-dimensional space to itself of full rank and is invertible; entry Q, at the same shape, has rank $4$ and is neither injective nor surjective.

Domains and codomains beyond column vectors

Several transformation entries have domains or codomains that are not spaces of column vectors, which is where the coordinatisation machinery earns its keep. Entry S maps $\mathbb{C}^3$ into the space of two-by-two matrices and has rank $2$ , so it is neither injective nor surjective. Entry T maps polynomials of degree at most four into polynomials of degree at most five by multiplication by $x - 2$ ; it is injective, and cannot be surjective because five dimensions cannot cover six. Entry U maps two-by-three matrices to $\mathbb{C}^4$ with rank $4$ , so it is surjective but cannot be injective from six dimensions into four. Entry V is a bijection between cubic polynomials and two-by-two matrices, with its inverse transformation given explicitly.

Using the catalogue as a source of counterexamples

The most demanding use of a designed catalogue is refutation. When a plausible converse is proposed — if the rank is less than the number of unknowns then the system is consistent, say — the catalogue is searched for an entry that satisfies the hypothesis and violates the conclusion. Entry E has rank $3$ against four unknowns and is inconsistent, which settles that particular converse in one line. Because each entry carries the full property list, the search is a table lookup rather than a computation, which is exactly what makes systematic converse-testing practical.

Selecting an entry for a given purpose

What kind of object is needed?A system of equations (A to J), a square matrix for its own sake (K, L), or a linear transformation (M to W).

Fix the shapeChoose square or rectangular, and whether equations should outnumber unknowns or the reverse. Overdetermined and underdetermined shapes behave differently and both are present.

Fix the solvabilityUnique solution, infinitely many, or none. For a controlled comparison choose a paired entry so that only this variable changes.

Fix the rank behaviourFull rank or deficient; nullity zero or positive. This determines the four subspaces and, for square entries, singularity and the determinant.

Fix the eigenstructure if relevantDistinct eigenvalues, repeated eigenvalues with matching geometric multiplicity, or a defective eigenvalue. Only the last blocks diagonalisation.

Read the property listEvery recorded quantity is exact, so the entry can be used directly as an expected-value reference without recomputation.

Equations

Entry A: system and reduced augmented matrix

EQ-ARCH-01

[\begin{matrix} 1 & - 1 & 2 & 1 \\ 2 & 1 & 1 & 8 \\ 1 & 1 & 0 & 5 \end{matrix}] ⟶ [\begin{matrix} 1 & 0 & 1 & 3 \\ 0 & 1 & - 1 & 2 \\ 0 & 0 & 0 & 0 \end{matrix}]

Three equations in three unknowns with $r = 2$ , $D = \{1, 2\}$ and $F = \{3, 4\}$ . Column $4$ is not a pivot column, so the system is consistent with one degree of freedom.

Entry A: vector form of the solution set

EQ-ARCH-02

[\begin{matrix} x_{1} \\ x_{2} \\ x_{3} \end{matrix}] = [\begin{matrix} 3 \\ 2 \\ 0 \end{matrix}] + x_{3} [\begin{matrix} - 1 \\ 1 \\ 1 \end{matrix}]

A particular solution plus a spanning vector for the null space of the coefficient matrix. Setting $x_3 = 1$ recovers the sample solution $(2, 3, 1)$ .

Entries D and E: identical coefficients, opposite outcomes

EQ-ARCH-03

[\begin{matrix} 1 & 0 & 3 & - 2 & 4 \\ 0 & 1 & 1 & - 3 & 0 \\ 0 & 0 & 0 & 0 & 0 \end{matrix}] [\begin{matrix} 1 & 0 & 3 & - 2 & 0 \\ 0 & 1 & 1 & - 3 & 0 \\ 0 & 0 & 0 & 0 & 1 \end{matrix}]

The reduced augmented matrices of D on the left and E on the right. Both coefficient parts are identical; only E has a leading one in the final column.

Shared null space of entries D and E

EQ-ARCH-04

N (A) = span ({[\begin{matrix} - 3 \\ - 1 \\ 1 \\ 0 \end{matrix}], [\begin{matrix} 2 \\ 3 \\ 0 \\ 1 \end{matrix}]}), rank (A) = 2, nullity (A) = 2

Every property that depends on the coefficient matrix alone is common to both entries. Only the constant vector distinguishes a consistent system from an inconsistent one.

Entry A: a defective eigenvalue

EQ-ARCH-05

α_{A} (0) = 2, γ_{A} (0) = 1, α_{A} (2) = 1, γ_{A} (2) = 1

The geometric multiplicity of the zero eigenvalue falls short of its algebraic multiplicity, so the coefficient matrix of entry A has no basis of eigenvectors and is not diagonalisable.

Entry B: a nonsingular contrast

EQ-ARCH-06

\det (B) = - 2, λ \in {- 1, 1, 2}, N (B) = {\vec{0}}

Three distinct eigenvalues force three independent eigenvectors, so entry B is diagonalisable; a non-zero determinant confirms nonsingularity independently.

Dimension constraints on the transformation entries

EQ-ARCH-07

\dim (U) > \dim (V) \Rightarrow T not injective, \dim (U) < \dim (V) \Rightarrow T not surjective

For $T \colon U \to V$ , relative dimension forbids one property outright. The transformation entries are chosen so that every cell the constraints leave open is occupied.

Rank and nullity across the catalogue

EQ-ARCH-08

rank (A) + nullity (A) = n

Holds for every entry and is the fastest consistency check on any recomputation. Entry I has $3 + 4 = 7$ and entry J has $4 + 5 = 9$ .

Variable Definitions

Symbols used on this page
Symbol	Name	Meaning	Domain / type
$m$	Row count	Number of equations, or the dimension of the codomain space	positive integer
$n$	Column count	Number of unknowns, or the dimension of the domain space	positive integer
$r$	Rank	Number of non-zero rows in the reduced form; recorded for every entry	0 to min(m,n)
$D$	Pivot column set	Indices of the columns containing leading ones	subset of 1..n+1
$Non-pivot column set$	Non-pivot column set	Complement of $D$ ; indexes the free variables	subset of 1..n+1
$A$	Coefficient matrix	The matrix analysed for each system entry, separately from its constant vector	m x n matrix
$\vec{b}$	Constant vector	The right-hand side; the only quantity distinguishing a paired entry from its partner	vector in C^m
$α_{A} (λ)$	Algebraic multiplicity	Multiplicity of $\lambda$ as a root of the characteristic polynomial	positive integer
$γ_{A} (λ)$	Geometric multiplicity	Dimension of the eigenspace; equals $\alpha$ for every eigenvalue exactly when the matrix is diagonalisable	positive integer

Worked Numerical Example

Problem statement

Use the paired entries D and E to establish, by direct computation rather than assertion, that the consistency of a linear system is determined by the pair $(A, \vec{b})$ and not by $A$ alone — and that everything else about the system's structure is determined by $A$ alone.

Confirm the coefficient matrices are identical
Entry D is the system $2x_1 + x_2 + 7x_3 - 7x_4 = 8$ , $-3x_1 + 4x_2 - 5x_3 - 6x_4 = -12$ , $x_1 + x_2 + 4x_3 - 5x_4 = 4$ . Entry E has the same left-hand sides with constants $2$ , $3$ and $2$ . The shared coefficient matrix is three by four.
$A = [\begin{matrix} 2 & 1 & 7 & - 7 \\ - 3 & 4 & - 5 & - 6 \\ 1 & 1 & 4 & - 5 \end{matrix}], {\vec{b}}_{D} = [\begin{matrix} 8 \\ - 12 \\ 4 \end{matrix}], {\vec{b}}_{E} = [\begin{matrix} 2 \\ 3 \\ 2 \end{matrix}]$
Reduce the coefficient matrix alone
Row reduction gives two non-zero rows, so $\operatorname{rank}(A) = 2$ and $\operatorname{nullity}(A) = 4 - 2 = 2$ . This computation uses no information about either constant vector, so its result applies to both entries without change.
$A ⟶ [\begin{matrix} 1 & 0 & 3 & - 2 \\ 0 & 1 & 1 & - 3 \\ 0 & 0 & 0 & 0 \end{matrix}], D = {1, 2}, F = {3, 4}$
Reduce entry D's augmented matrix
Carrying the constants through gives a final column of $4$ , $0$ , $0$ . The third row is entirely zero, representing the redundant equation $0 = 0$ , and column $5$ contains no leading one. The system is consistent.
$[\begin{matrix} 1 & 0 & 3 & - 2 & 4 \\ 0 & 1 & 1 & - 3 & 0 \\ 0 & 0 & 0 & 0 & 0 \end{matrix}], r = 2, 5 \notin D$
Reduce entry E's augmented matrix
The identical row operations on the identical coefficient part now leave a final column of $0$ , $0$ , $1$ . The third row reads $0 = 1$ , so no assignment of the unknowns satisfies it. Note that the rank of the augmented matrix is $3$ while the rank of $A$ remains $2$ .
$[\begin{matrix} 1 & 0 & 3 & - 2 & 0 \\ 0 & 1 & 1 & - 3 & 0 \\ 0 & 0 & 0 & 0 & 1 \end{matrix}], r = 3, 5 \in D$
Write entry D's solution set in vector form
The pivot rows give $x_1 = 4 - 3x_3 + 2x_4$ and $x_2 = -x_3 + 3x_4$ , with $x_3$ and $x_4$ free. Collecting coefficients gives a particular solution plus a two-parameter homogeneous part. Setting $x_3 = 2$ and $x_4 = 1$ recovers the sample solution $(0, 1, 2, 1)$ .
$\vec{x} = [\begin{matrix} 4 \\ 0 \\ 0 \\ 0 \end{matrix}] + x_{3} [\begin{matrix} - 3 \\ - 1 \\ 1 \\ 0 \end{matrix}] + x_{4} [\begin{matrix} 2 \\ 3 \\ 0 \\ 1 \end{matrix}]$
Observe what E retains despite having no solutions
Entry E has an empty solution set, but its associated homogeneous system is identical to D's and has the same two-dimensional solution space. Rank, nullity and all four fundamental subspaces are shared. Inconsistency removes the particular solution and nothing else.
$N (A) = span ({[\begin{matrix} - 3 \\ - 1 \\ 1 \\ 0 \end{matrix}], [\begin{matrix} 2 \\ 3 \\ 0 \\ 1 \end{matrix}]}) for both$
Cross-check with the second pair
Entries G and H repeat the experiment at five equations in two unknowns. Their shared coefficient matrix has rank $2$ and nullity $0$ , so consistency would force uniqueness. G is consistent with the single solution $(6, -2)$ ; H is inconsistent, its reduced augmented matrix carrying a leading one in column $3$ . A different shape, the same conclusion.
$Cross-check with the second pair$
State the conclusion the pairs establish
Consistency is not a property of the coefficient matrix. It is a property of whether $\vec{b}$ lies in the column space of $A$ , and the reduced augmented matrix is precisely the instrument that answers that question. Reducing $A$ by itself is sufficient for rank, nullity and every subspace, and insufficient for solvability.
$S \neq \emptyset \Leftrightarrow \vec{b} \in C (A) \Leftrightarrow rank ([A ∣ \vec{b}]) = rank (A)$

Result

Two entries sharing a coefficient matrix produced opposite solvability outcomes while agreeing on rank $2$ , nullity $2$ and every fundamental subspace. The general statement follows: solvability is decided by whether the constant vector lies in the column space, and the augmented reduction is the test. A second pair at an entirely different shape reproduced the result, which is the assurance a single example cannot give.

Applications & Industry Use

Numerical software engineering

A regression corpus with exact expected values

Because every entry is built from small integers and every recorded property is exact, the catalogue functions directly as a test corpus. A rank routine, a null space basis routine and an eigenvalue routine can each be checked against known answers, and entries with deliberately awkward structure — deficient rank, defective eigenvalues, inconsistency — exercise the branches that random well-conditioned test data never reaches.

Structural engineering

Recognising indeterminacy patterns before analysis

The nullity of an equilibrium matrix is the degree of static indeterminacy and the rank of the augmented matrix decides whether a given load case is equilibrium-compatible. Entries D and E model exactly that distinction: the structure is unchanged, but one load case is compatible and the other is not. Having a worked instance of the pattern in hand shortens the diagnosis of a real model that will not solve.

Control systems

Rank-deficient and defective examples for algorithm validation

Controllability and observability tests, pole placement and modal decomposition all behave differently when a rank is deficient or an eigenvalue is defective. Entry A supplies a defective eigenvalue with algebraic multiplicity two and geometric multiplicity one, and entries K and L supply repeated eigenvalues that are not defective, which together distinguish an implementation that handles multiplicity correctly from one that only appears to.

Chemical process engineering

Feasible and infeasible specifications on one flowsheet

A blend or reaction balance is a linear system whose coefficient matrix encodes the flowsheet and whose constant vector encodes the specification. The paired entries model the situation in which the same plant meets one specification and cannot meet another, showing that the diagnosis belongs to the specification and that redesigning the plant is not the indicated response.

Education technology and assessment

Item banks with known structural answers

Automatically generated problems need reference answers that are certain, not merely computed. A catalogue whose entries are exact and whose property lists are complete supplies verified answers for every question that can be asked about a given instance, including the awkward ones about multiplicity and diagonalisability.

Computational mathematics

Systematic testing of proposed converses

Refuting a plausible-sounding converse requires an object satisfying the hypothesis and failing the conclusion. Because each entry carries a complete property list, converse-testing becomes a table search rather than a search over matrices, and a single entry frequently settles a question that would otherwise absorb an afternoon.

Design Considerations

Use a pair, not an example, to attribute cause

A single instance shows that a phenomenon can occur; it cannot show what produces it. Changing exactly one feature and observing what changes is the only reliable attribution, which is why the paired entries do disproportionate work. When constructing an example of your own, construct its partner as well.

Do not generalise a pattern from small integer entries

Entries are built from small integers so that every property is exact and hand-verifiable. That choice introduces incidental structure — integer eigenvalues, factorable characteristic polynomials, exact pivots — that is not typical. Patterns observed across the catalogue are hypotheses to be proved, never conclusions, and floating-point behaviour in particular cannot be inferred from them.

Check which properties an entry legitimately has

Determinants, inverses and eigenvalues require a square matrix; injectivity and surjectivity require a transformation. An entry's property list omits what does not apply rather than recording a placeholder, and reading a value that is absent as though it were zero is a category error rather than an arithmetic one.

Prefer an awkward entry for a regression test

Well-conditioned square nonsingular examples pass almost any implementation. The valuable test cases are the deficient ones: an inconsistent system, a rank-deficient rectangular matrix, a defective eigenvalue, a transformation whose domain and codomain differ in dimension. Building a test suite from the awkward entries first finds defects that a suite of tidy ones never will.

Record the full property list when adding an entry

The value of the catalogue comes from every entry being worked to the same schedule, which is what makes cross-entry comparison and table lookup possible. An entry recorded partially is not a smaller contribution but a different kind of object, and it silently breaks the assumption that any question can be answered by lookup.

Treat an entry as evidence, never as proof

An instance can refute a universal claim and can motivate one, but it cannot establish one. The catalogue is at its most valuable in the refutation role, where a single entry is logically decisive, and at its least valuable when a pattern across several entries is mistaken for a demonstration.

Standards & Codes

Notation, interchange and numerical standards that govern how this material is written down, stored and computed in production systems.

Applicable standards, conventions and reference implementations
Reference	Title	Relevance to this topic
`ISO/IEC/IEEE 29119`	Software and systems engineering — Software testing	Frames the catalogue's role as a test basis with documented expected results, and the paired entries as boundary and equivalence-partition cases separating consistent from inconsistent behaviour.
`Matrix Market exchange formats`	NIST Matrix Market file formats for matrix data	The de facto plain-text formats for interchanging test matrices, including coordinate form for sparse data and explicit symmetry qualifiers, in which a catalogue of this kind would be published for machine consumption.
`SuiteSparse Matrix Collection`	Curated collection of sparse matrices from applications	The large-scale analogue of this catalogue: a curated, versioned corpus with recorded properties, used as the standard benchmark set for sparse solvers and demonstrating the value of designed rather than random test data.
`ISO 80000-2`	Quantities and units — Part 2: Mathematics	Fixes the notation in which every property list is recorded, so that entries transcribed into a document or a test fixture remain unambiguous.
`IEEE 754-2019`	IEEE Standard for Floating-Point Arithmetic	Explains why the exact integer entries must be converted with care when used as floating-point test data: rank and multiplicity results that are exact over the integers become tolerance-dependent once rounded.

Material Selection

For a mathematical topic, "material" is the numeric representation: the scalar field, storage format and precision the computation is built from.

Representation and precision selection
Representation	Select when	Trade-off
Small integer entries, exact arithmetic	The design choice throughout this catalogue, so that every recorded property can be verified by hand and reproduced exactly.	Every value is certain and independently checkable, but the entries are unrepresentatively well behaved and say nothing about conditioning.
Exact rational arithmetic	Reproducing an entry's property list in a computer algebra system, where reduction introduces fractions.	Preserves exactness through row reduction and eigenvector computation, at the cost of coefficient growth that is invisible in the printed entry but real in the computation.
IEEE 754 binary64 conversion of an entry	Using the catalogue as input to a numerical library under test.	Small integers convert exactly, so the input is faithful, but derived quantities such as rank and multiplicity become tolerance decisions and the exact expected values must be compared with a stated tolerance.
Deliberately ill-conditioned companion data	Testing whether an implementation degrades gracefully, which the exact entries cannot reveal.	Exposes tolerance and pivoting behaviour that integer data hides, but expected values are then themselves approximate and the test becomes a comparison of two estimates.
Modular arithmetic over a prime	Independently certifying a rank or determinant recorded in a property list.	Fast and exact with no growth, but a rank can differ modulo an unlucky prime, so agreement across several primes is needed before the check counts.
Dense storage for every entry	The natural choice here: entries are small and mostly full.	Simple and fast at these sizes, but the catalogue therefore exercises none of the fill-in, ordering and sparsity behaviour that dominates large-scale practice.

Manufacturing Notes

Implementation notes — how the result is actually produced by hand, by algorithm and by library, including cost and numerical behaviour.

Reproducing a property list from scratch

Work in a fixed order, because later properties depend on earlier ones. Reduce the augmented matrix and record $r$ , $D$ and $F$ ; decide consistency; write the vector form of the solution set; reduce the coefficient matrix alone; extract bases for the four subspaces; then, if the matrix is square, compute the determinant, the characteristic polynomial, the eigenvalues and each eigenspace. Checking $\operatorname{rank} + \operatorname{nullity} = n$ after the subspace step catches most arithmetic slips before they propagate.

Two independent routes to the column space

A column space basis can be taken as the original columns indexed by $D$ , or obtained by reducing the transpose and reading its non-zero rows as columns. The two produce different-looking bases for the same subspace, and agreement of their dimensions is a free check. A third route, through the extended echelon form, yields the same subspace as the null space of a small auxiliary matrix and validates the other two.

Cost of the full property list

For an entry of size $n$ the reductions cost $O(n^3)$ , the four subspace bases come almost free from reductions already performed, and the determinant is a by-product of elimination. The expensive part is the eigenstructure: factoring the characteristic polynomial exactly is only feasible because the entries were constructed to have integer or small rational eigenvalues, which is the main reason the catalogue is built from designed rather than random data.

Using entries as fixtures in an automated test suite

Encode each entry once with its expected property list and drive every relevant routine from it. Compare exact integer results with equality and floating-point results against a tolerance scaled to the matrix norm. Assert dimensions and subspace membership rather than specific basis vectors, since a correct implementation may legitimately return a different basis for the same subspace.

Verifying a claimed eigenspace basis

Do not compare basis vectors between two computations; compare subspaces. Confirm that each claimed vector satisfies $A\vec{x} = \lambda \vec{x}$ exactly, then confirm that the count of independent vectors equals the nullity of $A - \lambda I_n$ . Together these establish that the reported set is a basis of the eigenspace without requiring it to match any particular reference list.

Failure Modes & Common Mistakes

Failure modes, root causes and prevention
Failure mode / mistake	Impact	Root cause	Prevention & detection
Generalising from a single entry	high	Observing a property in one worked instance and treating it as a general theorem.	An instance can refute a universal claim but never establish one. Use an entry to form a conjecture, then prove it or find the entry that refutes it.
Confusing the members of a pair	medium	Quoting entry D's solution set for entry E, or vice versa, because the coefficient matrices are identical.	The distinguishing feature of a pair is the one thing that must be checked at every use. Record the constant vector alongside any citation of a paired entry.
Assuming integer eigenvalues are typical	medium	Inferring from a catalogue of designed entries that characteristic polynomials generally factor over the rationals.	Integer spectra are a construction convenience. A random integer matrix almost never has them, and an implementation tuned on such data will mishandle irrational and complex spectra.
Reducing the coefficient matrix when consistency is the question	high	Analysing $A$ alone and reporting a solution set, discarding the constant vector that decides solvability.	Reduce the augmented matrix. Entries D and E have identical coefficient reductions and opposite outcomes, which is the standing demonstration of the point.
Reading an absent property as zero	medium	Treating a rectangular entry as having a determinant of zero because no determinant is recorded.	Determinant, inverse and eigenvalues are undefined for non-square matrices. An omitted property means inapplicable, not zero.
Comparing eigenvector lists instead of eigenspaces	medium	Declaring an implementation wrong because it returned different basis vectors for an eigenspace of dimension two or more.	Verify that each returned vector satisfies the eigenvector equation and that the count matches the geometric multiplicity. Any basis of the correct subspace is correct.
Using an exact entry to draw conclusions about conditioning	medium	Concluding that an algorithm is numerically sound because it handled small integer data correctly.	Exact entries test correctness of logic, not stability. Pair them with deliberately ill-conditioned data before making any claim about numerical behaviour.
Assuming repeated eigenvalues block diagonalisation	medium	Generalising from entry A, whose repeated eigenvalue is defective, to entries K and L, whose repeated eigenvalues are not.	Compare $\gamma_A(\lambda)$ with $\alpha_A(\lambda)$ for each eigenvalue individually. Repetition alone decides nothing; only a shortfall in geometric multiplicity does.
Treating an overdetermined system as automatically inconsistent	medium	Assuming that more equations than unknowns forces a contradiction.	Entry G has five equations in two unknowns and is consistent with a unique solution, three equations being redundant. Entry H, at the same shape, is inconsistent. Shape alone decides nothing.

FAQs

Why are two entries built on the same coefficient matrix?

So that a single variable can be isolated. Entries D and E differ only in the constant vector, and entries G and H do the same at a different size. Because everything else is held fixed, any difference in behaviour must be attributable to the constant vector, which is how the catalogue demonstrates that consistency belongs to the pair $(A, \vec{b})$ rather than to $A$ .

Can an entry be used as proof of a general statement?

No, with one exception. An instance cannot establish a universal claim however many entries agree. It can, however, refute one outright: a single entry satisfying the hypothesis and failing the conclusion settles the matter. The catalogue is therefore a proof instrument only in the negative direction, and a source of conjectures in the positive one.

Why are the entries larger than the examples usually shown in a discussion?

Because small examples reinforce patterns that do not survive. At size two or three, pivots tend to fall on the diagonal, rank tends to be full, and repeated eigenvalues are rare. Entries of size four and above, some rectangular and some badly deficient, expose behaviour that a two-by-two example structurally cannot exhibit.

How should the catalogue be used to test a numerical library?

Encode each entry with its exact expected property list, then drive rank, null space, column space, determinant and eigenvalue routines from it. Prefer the awkward entries: inconsistent systems, rank-deficient rectangles and defective eigenvalues exercise the branches that well-conditioned random data never reaches. Compare subspaces rather than specific basis vectors.

Are the small integer entries realistic?

Not in their conditioning, and deliberately so. Exact integers make every property hand-verifiable and every expected value certain, which is what a reference corpus requires. They say nothing about rounding, pivoting or tolerance behaviour, so a numerical assessment needs ill-conditioned companion data alongside them.

What does it mean that some entries have properties omitted rather than recorded as zero?

It means the property does not apply. Determinants, inverses and eigenvalues are defined only for square matrices, and injectivity and surjectivity only for transformations. Which properties are absent is itself information about the entry, and reading an absence as a value is a category error.

Why do so many transformation entries exist relative to systems and matrices?

Because the transformation entries must cover a two-dimensional grid: injective or not, surjective or not, at each of three relative dimension relationships between domain and codomain. Relative dimension forbids some cells outright, and the entries are chosen so that every cell it leaves open is actually occupied.

References

Beezer, R. A. A First Course in Linear Algebra, Version 0.70. University of Puget Sound, 2006. Chapter A, Archetypes A through W. Licensed under the GNU Free Documentation License v1.2.
ISO 80000-2:2019, Quantities and units — Part 2: Mathematics. International Organization for Standardization.
Boisvert, R. F., Pozo, R., Remington, K., Barrett, R. and Dongarra, J. The Matrix Market: A Web Resource for Test Matrix Collections. Chapman and Hall, 1997.
Davis, T. A. and Hu, Y. The University of Florida Sparse Matrix Collection. ACM Transactions on Mathematical Software, volume 38, 2011.
ISO/IEC/IEEE 29119-4:2021, Software and systems engineering — Software testing — Part 4: Test techniques.
Higham, N. J. Accuracy and Stability of Numerical Algorithms, 2nd edition. Society for Industrial and Applied Mathematics, 2002. On the limitations of well-conditioned test data.

AI Suggested Questions

Construct a partner for a given 4-by-6 system that shares its coefficient matrix but is inconsistent, and show the reduced augmented matrices side by side.
Which entries in the catalogue would refute the claim that a system with fewer equations than unknowns is always consistent?
Design a test suite from the catalogue that exercises every branch of a rank-revealing factorisation, and justify each entry chosen.
Take an entry with a defective eigenvalue and show explicitly why no basis of eigenvectors can exist for it.
Compare the two routes to a column space basis on a rank-deficient entry, and prove the resulting bases span the same subspace.
What structural properties of a small integer matrix are atypical, and how would a randomly generated integer matrix of the same size differ?

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