Anti-Diagonal Matrix¶

In mathematics, an anti-diagonal matrix is a square matrix where all the entries are zero except those on the diagonal going from the lower left corner to the upper right corner (↗) , known as the anti-diagonal (sometimes Harrison diagonal, secondary diagonal, trailing diagonal, minor diagonal, off diagonal or bad diagonal).

Version: v1 · 2026-09-28 · History
Domain-specific #: 7971
Domain group: Formal Sciences
Origin domain: Mathematics
Subdomains: Linear Algebra, Matrix Theory → Mathematics

Core Idea¶

Anti-Diagonal Matrix is treated here as the recurring mathematics, logic, and statistics identity summarized by this source-grounded definition: In mathematics, an anti-diagonal matrix is a square matrix where all the entries are zero except those on the diagonal going from the lower left corner to the upper right corner (↗) , known as the anti-diagonal (sometimes Harrison diagonal, secondary diagonal, trailing diagonal, minor diagonal, off diagonal or bad diagonal).

In mathematics, an anti-diagonal matrix is a square matrix where all the entries are zero except those on the diagonal going from the lower left corner to the upper right corner (↗) , known as the anti-diagonal (sometimes Harrison diagonal, secondary diagonal, trailing diagonal, minor diagonal, off diagonal or bad diagonal). An anti-diagonal matrix is invertible if and only if the entries on the diagonal from the lower left corner to the upper right corner are nonzero. An matrix is an anti-diagonal matrix if the th element is zero for all rows and columns whose indices do not sum to .

Furthermore, the product of an anti-diagonal matrix with a diagonal matrix is anti-diagonal, as is the product of a diagonal matrix with an anti-diagonal matrix. The inverse of an invertible anti-diagonal matrix is also anti-diagonal, as can be seen from the paragraph above. However, the sign of this determinant varies because the one nonzero signed elementary product from an anti-diagonal matrix has a different sign depending on whether the permutation related to it is odd or even.

For Anti-Diagonal Matrix, the abstraction is narrower than the article's general subject matter: a positive case must preserve In mathematics, an anti-diagonal matrix is a square matrix where all the entries are zero except those on the diagonal going from the lower left corner to the upper right corner (↗) , known as the anti-diagonal (sometimes Harrison diagonal, secondary diagonal, trailing diagonal, minor diagonal, off diagonal or bad diagonal). Retaining only the name, a familiar example, or a downstream effect is insufficient. The specialist roles and tests remain anchored in mathematics, logic, and statistics, which is why this identity is domain-specific rather than prime.

How would you explain it like I'm…

The Slanty-Line Number Grid

A matrix is a square grid of numbers. In an anti-diagonal matrix, every number is zero except the ones on the slanted line that goes from the bottom-left corner up to the top-right corner.

The Backward-Diagonal Grid

A matrix is a square grid of numbers arranged in rows and columns. The main diagonal goes from the top-left corner to the bottom-right. The anti-diagonal goes the other way, from the bottom-left corner to the top-right. An anti-diagonal matrix has zeros everywhere except possibly on that anti-diagonal. It can be 'undone' (it has an inverse) exactly when none of the numbers on the anti-diagonal are zero.

Secondary-Diagonal-Only Matrix

An anti-diagonal matrix is a square matrix whose entries are all zero except on the anti-diagonal, the line from the lower-left corner to the upper-right corner (also called the secondary or minor diagonal, among other names). For an n×n matrix with rows and columns numbered 1 to n, that means entry (i, j) is zero unless i + j = n + 1. It is invertible if and only if all the anti-diagonal entries are nonzero, and its inverse is also anti-diagonal. Multiplying an anti-diagonal matrix by a diagonal matrix, on either side, gives another anti-diagonal matrix. Its determinant is plus or minus the product of the anti-diagonal entries, with the sign depending on the size of the matrix.

An anti-diagonal matrix is an n×n matrix A with a_ij = 0 whenever i + j ≠ n + 1, so nonzero entries can occur only on the anti-diagonal running from the lower-left to the upper-right corner (also called the secondary, minor, trailing, or Harrison diagonal). It is invertible if and only if every anti-diagonal entry is nonzero, in which case its inverse is again anti-diagonal. The product of an anti-diagonal matrix and a diagonal matrix, in either order, is anti-diagonal. In the Leibniz expansion of the determinant, only one elementary product is nonzero — the one corresponding to the order-reversing permutation — so the determinant is the product of the anti-diagonal entries times the sign of that permutation, which is odd or even depending on n.

Structural Signature¶

Sig role-phrases:

Defining carrier — which can be used to reverse the elements of an array (as a column matrix) by multiplying on the left.
Constitutive relation — The determinant of an anti-diagonal matrix has absolute value given by the product of the entries on the diagonal from the lower left corner to the upper right corner.
Operating condition — An matrix is an anti-diagonal matrix if the th element is zero for all rows and columns whose indices do not sum to .
Recognition evidence — a_{ij} = 0 \forall i,j \in \left{1, \ldots, n\right}, (i+j \ne n+1).
Admissible variation — Furthermore, the product of an anti-diagonal matrix with a diagonal matrix is anti-diagonal, as is the product of a diagonal matrix with an anti-diagonal matrix.
Characteristic consequence — An anti-diagonal matrix is invertible if and only if the entries on the diagonal from the lower left corner to the upper right corner are nonzero.
Failure boundary — The inverse of an invertible anti-diagonal matrix is also anti-diagonal, as can be seen from the paragraph above.

What It Is Not¶

Not the whole field of mathematics, logic, and statistics. The node requires the specific identity stated by In mathematics, an anti-diagonal matrix is a square matrix where all the entries are zero except those on the diagonal going from the lower left corner to the upper right corner (↗) , known as the anti-diagonal (sometimes Harrison diagonal, secondary diagonal, trailing diagonal, minor diagonal, off diagonal or bad diagonal).
Not an over-broad reading. However, the sign of this determinant varies because the one nonzero signed elementary product from an anti-diagonal matrix has a different sign depending on whether the permutation related to it is odd or even.
Not an over-broad reading. An matrix is an anti-diagonal matrix if the th element is zero for all rows and columns whose indices do not sum to .
Not an over-broad reading. In mathematics, an anti-diagonal matrix is a square matrix where all the entries are zero except those on the diagonal going from the lower left corner to the upper right corner (↗) , known as the anti-diagonal (sometimes Harrison diagonal, secondary diagonal, trailing diagonal, minor diagonal, off diagonal or bad diagonal).
Not automatically Diagonal Matrix. Retrieval proximity does not establish equivalence; the two identities must be compared by carrier, operation, and failure boundary.

Scope of Application¶

Anti-Diagonal Matrix applies literally inside mathematics, logic, and statistics wherever the source-defined carrier and relation can be established. Its documented habitats include:

1 & 0 & 0 & 0 & 0. which can be used to reverse the elements of an array (as a column matrix) by multiplying on the left.
Formal definition. An matrix is an anti-diagonal matrix if the th element is zero for all rows and columns whose indices do not sum to .
Formal definition. a_{ij} = 0 \forall i,j \in \left{1, \ldots, n\right}, (i+j \ne n+1).
Properties. Furthermore, the product of an anti-diagonal matrix with a diagonal matrix is anti-diagonal, as is the product of a diagonal matrix with an anti-diagonal matrix.
Properties. An anti-diagonal matrix is invertible if and only if the entries on the diagonal from the lower left corner to the upper right corner are nonzero.
Properties. The inverse of an invertible anti-diagonal matrix is also anti-diagonal, as can be seen from the paragraph above.

Outside mathematics, logic, and statistics, the name should be retained only when these same operational conditions survive; otherwise the comparison belongs to the broader parent Pattern or should be marked as analogy.

Clarity¶

A clear use of Anti-Diagonal Matrix names the carrier, the operative relation, and the conditions under which the source treats the identity as present. The minimal definition is In mathematics, an anti-diagonal matrix is a square matrix where all the entries are zero except those on the diagonal going from the lower left corner to the upper right corner (↗) , known as the anti-diagonal (sometimes Harrison diagonal, secondary diagonal, trailing diagonal, minor diagonal, off diagonal or bad diagonal). The strongest recognition evidence in the frozen account is: a_{ij} = 0 \forall i,j \in \left{1, \ldots, n\right}, (i+j \ne n+1). A report should distinguish that evidence from a proxy, consequence, or common implementation. It should also state the qualification However, the sign of this determinant varies because the one nonzero signed elementary product from an anti-diagonal matrix has a different sign depending on whether the permutation related to it is odd or even. so that a reader can reproduce the classification rather than infer it from topical resemblance.

Manages Complexity¶

Anti-Diagonal Matrix compresses multiple mathematics, logic, and statistics details into a stable diagnostic relation. The source shows both the central mechanism—the determinant of an anti-diagonal matrix has absolute value given by the product of the entries on the diagonal from the lower left corner to the upper right corner.—and the practical consequence—an anti-diagonal matrix is invertible if and only if the entries on the diagonal from the lower left corner to the upper right corner are nonzero. This compression makes cases comparable while leaving parameters, conventions, exceptions, and evidential quality explicit. It is lossy by design: local history and implementation details may be omitted only when they do not alter the defining relation.

Abstract Reasoning¶

Type the carrier. Identify the mathematics, logic, and statistics entities to which the claim applies.
State the relation. Use the source-grounded identity: In mathematics, an anti-diagonal matrix is a square matrix where all the entries are zero except those on the diagonal going from the lower left corner to the upper right corner (↗) , known as the anti-diagonal (sometimes Harrison diagonal, secondary diagonal, trailing diagonal, minor diagonal, off diagonal or bad diagonal).
Check operation and conditions. An matrix is an anti-diagonal matrix if the th element is zero for all rows and columns whose indices do not sum to .
Demand recognition evidence. a_{ij} = 0 \forall i,j \in \left{1, \ldots, n\right}, (i+j \ne n+1).
Test variation. Change an implementation or setting while preserving furthermore, the product of an anti-diagonal matrix with a diagonal matrix is anti-diagonal, as is the product of a diagonal matrix with an anti-diagonal matrix.
Run the collapse test. Remove the defining operation; if the label still seems equally apt, only a topic or correlate was retained.
Reduce cautiously. When the specialist conditions cannot be carried, route the residual comparison to Pattern.

Knowledge Transfer¶

Within the home domain. Knowledge about Anti-Diagonal Matrix transfers literally when a new case preserves the same carrier type, relation, and recognition test. which can be used to reverse the elements of an array (as a column matrix) by multiplying on the left. An matrix is an anti-diagonal matrix if the th element is zero for all rows and columns whose indices do not sum to .

Beyond the home domain. No canonical parent is asserted for Anti-Diagonal Matrix. An outside case receives the specialist name only when the same typed roles and rejection conditions can be filled literally; otherwise the comparison remains an analogy pending later graph densification.

Examples¶

Canonical¶

An matrix is an anti-diagonal matrix if the th element is zero for all rows and columns whose indices do not sum to . This case is canonical because it supplies a concrete carrier and lets the defining relation be checked rather than merely named.

Mapped back: carrier → the entities in the documented case; operation → In mathematics, an anti-diagonal matrix is a square matrix where all the entries are zero except those on the diagonal going from the lower left corner to the upper right corner (↗) , known as the anti-diagonal (sometimes Harrison diagonal, secondary diagonal, trailing diagonal, minor diagonal, off diagonal or bad diagonal); recognition evidence → a_{ij} = 0 \forall i,j \in \left{1, \ldots, n\right}, (i+j \ne n+1)

Applied / In Practice¶

a_{ij} = 0 \forall i,j \in \left{1, \ldots, n\right}, (i+j \ne n+1). The applied case shows how the identity is used under a second setting or qualification while keeping the same operative relation.

Mapped back: changed setting → Formal definition; invariant → In mathematics, an anti-diagonal matrix is a square matrix where all the entries are zero except those on the diagonal going from the lower left corner to the upper right corner (↗) , known as the anti-diagonal (sometimes Harrison diagonal, secondary diagonal, trailing diagonal, minor diagonal, off diagonal or bad diagonal); boundary → the case exits the class when however, the sign of this determinant varies because the one nonzero signed elementary product from an anti-diagonal matrix has a different sign depending on whether the permutation related to it is odd or even

Structural Tensions¶

T1 — Stable identity versus admissible variation. However, the sign of this determinant varies because the one nonzero signed elementary product from an anti-diagonal matrix has a different sign depending on whether the permutation related to it is odd or even. The tension matters because emphasizing only one side either dissolves the identity or overstates what the evidence and domain conventions warrant.

Diagnostic: Which changes preserve the defining relation, and which replace it?

T2 — Recognition versus proxy. An matrix is an anti-diagonal matrix if the th element is zero for all rows and columns whose indices do not sum to . The tension matters because emphasizing only one side either dissolves the identity or overstates what the evidence and domain conventions warrant.

Diagnostic: Does the cited evidence establish the identity or only a correlated sign?

T3 — Definition versus implementation. In mathematics, an anti-diagonal matrix is a square matrix where all the entries are zero except those on the diagonal going from the lower left corner to the upper right corner (↗) , known as the anti-diagonal (sometimes Harrison diagonal, secondary diagonal, trailing diagonal, minor diagonal, off diagonal or bad diagonal). The tension matters because emphasizing only one side either dissolves the identity or overstates what the evidence and domain conventions warrant.

Diagnostic: Is the observed implementation constitutive, optional, or merely common?

T4 — Scope versus overextension. a_{ij} = 0 \forall i,j \in \left{1, \ldots, n\right}, (i+j \ne n+1). The tension matters because emphasizing only one side either dissolves the identity or overstates what the evidence and domain conventions warrant.

Diagnostic: Can every claimed application fill the same typed roles without metaphor?

T5 — Transfer versus domain accent. which can be used to reverse the elements of an array (as a column matrix) by multiplying on the left. The tension matters because emphasizing only one side either dissolves the identity or overstates what the evidence and domain conventions warrant.

Diagnostic: Does the receiving case instantiate Anti-Diagonal Matrix literally, co-instantiate Pattern, or only resemble it?

T6 — Autonomy versus reduction. The determinant of an anti-diagonal matrix has absolute value given by the product of the entries on the diagonal from the lower left corner to the upper right corner. The tension matters because emphasizing only one side either dissolves the identity or overstates what the evidence and domain conventions warrant.

Diagnostic: What does Anti-Diagonal Matrix distinguish that the broader parent Pattern leaves together?

Structural–Framed Character¶

Anti-Diagonal Matrix is structural-leaning. Its structural side is the repeatable organization summarized by In mathematics, an anti-diagonal matrix is a square matrix where all the entries are zero except those on the diagonal going from the lower left corner to the upper right corner (↗) , known as the anti-diagonal (sometimes Harrison diagonal, secondary diagonal, trailing diagonal, minor diagonal, off diagonal or bad diagonal). Its framed side is the mathematics, logic, and statistics vocabulary that fixes the carrier, evidence, exceptions, and admissible transformations.

Evaluative weight: the identity can be stated descriptively even when applications carry practical stakes. Human-practice dependence: the source-grounded carrier determines whether the relation exists independently or is constituted by a practice. Institutional origin: disciplinary conventions stabilize the name and test. Vocabulary portability: An matrix is an anti-diagonal matrix if the th element is zero for all rows and columns whose indices do not sum to . Import versus recognition: literal transfer requires the same mechanism; shape alone is analogy.

Its portable skeleton is Pattern. Its character: a recurring specialist identity whose thin organization can be abstracted, while its operational meaning remains domain-bound.

Structural Core vs. Domain Accent¶

What is skeletal. In mathematics, an anti-diagonal matrix is a square matrix where all the entries are zero except those on the diagonal going from the lower left corner to the upper right corner (↗) , known as the anti-diagonal (sometimes Harrison diagonal, secondary diagonal, trailing diagonal, minor diagonal, off diagonal or bad diagonal). The stable skeleton is the typed relation expressed in that definition and the entry's recognition and collapse tests. The source identifies these operative conditions: which can be used to reverse the elements of an array (as a column matrix) by multiplying on the left. The determinant of an anti-diagonal matrix has absolute value given by the product of the entries on the diagonal from the lower left corner to the upper right corner. It further constrains recognition and variation through: An matrix is an anti-diagonal matrix if the th element is zero for all rows and columns whose indices do not sum to . a{ij} = 0 \forall i,j \in \left{1, \ldots, n\right}, (i+j \ne n+1).

What is domain-bound. mathematics, logic, and statistics supplies the operative entities, technical vocabulary, warrants, and exceptions that make Anti-Diagonal Matrix literal. Its documented scope includes the condition that which can be used to reverse the elements of an array (as a column matrix) by multiplying on the left. Another bounded application condition is that An matrix is an anti-diagonal matrix if the th element is zero for all rows and columns whose indices do not sum to . These are not decorative examples; they determine which carrier and evidence can fill the abstraction's roles.

Why no parent is asserted. Removing those specialist details does not currently yield one live catalog node that is a necessary genus for every instance. The entry is therefore approved as unparented rather than attached by topical resemblance. Its collapse evidence remains specific—Furthermore, the product of an anti-diagonal matrix with a diagonal matrix is anti-diagonal, as is the product of a diagonal matrix with an anti-diagonal matrix.—and future graph densification may discover a defensible relation only if it preserves that boundary.

This entry is a kind of Matrix.

Approved unparented node. No current live node supplies a defensible necessary genus or structural prerequisite for Anti-Diagonal Matrix. The reviewed identity is: In mathematics, an anti-diagonal matrix is a square matrix where all the entries are zero except those on the diagonal going from the lower left corner to the upper right corner (↗), known as the anti-diagonal (sometimes Harrison diagonal, secondary diagonal, trailing diagonal, minor diagonal, off diagonal or bad diagonal). The accelerated suggestion was declined because topical or lexical similarity does not establish hierarchy; the node is admitted without a parent pending later graph densification.
Related reasoning operations. Evidence, representation, comparison, classification, transformation, or evaluation may participate in particular cases, but participation does not make any one of them a necessary parent of every instance.

Relationships to Other Abstractions¶

Current abstraction Anti-Diagonal Matrix Domain-specific

Parents (1) — more general patterns this builds on

Anti-Diagonal Matrix is a kind of Matrix Domain-specific

An anti-diagonal matrix is a square matrix constrained to vanish away from its anti-diagonal.

Hierarchy paths (5) — routes to 5 parentless roots

Anti-Diagonal Matrix → Matrix → Tensor → Transformation → Function (Mapping)

Show alternative paths (4)

Neighborhood in Abstraction Space¶

Anti-Diagonal Matrix sits in a moderately populated region (47^th percentile for distinctiveness): it has near-neighbors but no dense thicket of look-alikes.

Family — Matrix Structures & Matroids (10 abstractions)

Nearest neighbors

S-procedure — 0.88
Integer matrix — 0.88
G-Matrix — 0.87
Hat matrix — 0.86
Pentadiagonal Matrix — 0.86

Computed from structural-signature embeddings · 2026-10-08

Not to Be Confused With¶

Pattern. The parent omits the specialist differentia. Tell: Can the case establish In mathematics, an anti-diagonal matrix is a square matrix where all the entries are zero except those on the diagonal going from the lower left corner to the upper right corner (↗) , known as the anti-diagonal (sometimes Harrison diagonal, secondary diagonal, trailing diagonal, minor diagonal, off diagonal or bad diagonal)?
Diagonal Matrix. A matrix whose off-main-diagonal entries are zero, so its coordinate axes decouple and addition, multiplication, inversion, powers, determinants, and spectral action reduce to scalar operations on diagonal entries. Tell: Which entry's carrier, operation, and failure condition are satisfied?
L-Matrix. A real square matrix with every diagonal entry strictly positive and every off-diagonal entry nonpositive—the positive-diagonal subclass of Z-matrices and the sign-pattern counterpart of a negated Metzler matrix. Tell: Which entry's carrier, operation, and failure condition are satisfied?
Bidiagonal matrix. A banded matrix whose potentially nonzero entries lie only on the main diagonal and one adjacent superdiagonal or subdiagonal, yielding simple determinants, eigenvalues, products, and efficient structured computations. Tell: Which entry's carrier, operation, and failure condition are satisfied?
A measurement, proxy, or consequence. Those may provide evidence without being the identity. Tell: Would Anti-Diagonal Matrix remain present if the detector or downstream effect changed?
A metaphorical analogue. A similar shape outside mathematics, logic, and statistics lacks the specialist mechanism. Tell: Do the native roles transfer literally, or only the parent Pattern?

References¶

Frozen Wikipedia discovery revision: https://en.wikipedia.org/wiki/Anti-diagonal_matrix (revision 1364626952).

The frozen Wikipedia revision is discovery provenance. The retained source set was reviewed for identity, formal or operational relation, and scope. The encyclopedia's structural synthesis is bounded to those claims; a thin authority surface is recorded as a nonblocking source-strengthening repair rather than concealed.