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.

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.

Scope of Application

• 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.

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.

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.

Abstract Reasoning

1. Type the carrier. Identify the mathematics, logic, and statistics entities to which the claim applies.
2. 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).
3. Check operation and conditions.

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.

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