# Antidiagonal vs. Diagonal: Know the Difference

By Shumaila Saeed || Updated on December 29, 2023

**The antidiagonal connects the opposite corners in a square matrix, moving from top-right to bottom-left, while the diagonal connects the corners in a matrix, moving from top-left to bottom-right.**

## Key Differences

Antidiagonal lines in a matrix move from the top-right corner to the bottom-left corner. This contrasts with diagonal lines, which typically run from the top-left corner to the bottom-right corner.

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In a square matrix, the antidiagonal elements are those where the row number plus the column number equals the size of the matrix plus one. On the other hand, diagonal elements are those where the row and column numbers are equal.

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The antidiagonal often represents a kind of symmetry that is opposite to the main diagonal in a matrix or a square. In contrast, the diagonal is usually seen as the primary line of symmetry, especially in square matrices.

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Visually, antidiagonal lines can be seen in patterns or designs that require a cross-sectional symmetry from top-right to bottom-left. Diagonal lines are more common and are recognized in patterns that need a straightforward left-to-right downward slant.

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In mathematics, antidiagonal matrices have specific properties and uses, particularly in advanced algebra and geometry, distinct from those of diagonal matrices, which are often used in calculus and linear algebra for simplifying equations.

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## Comparison Chart

### Direction in a Matrix

Moves from top-right to bottom-left

Moves from top-left to bottom-right

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### Element Positions

Row number + Column number equals size of matrix + 1

Row number equals column number

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### Symmetry in Matrices

Represents opposite symmetry to the main diagonal

Often the primary line of symmetry

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### Visual Representation

Seen in cross-sectional top-right to bottom-left lines

Commonly recognized in left-to-right downward patterns

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### Mathematical Use

Used in advanced algebra and geometry

Used in calculus and linear algebra

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## Antidiagonal and Diagonal Definitions

#### Antidiagonal

Elements of a matrix that lie on the antidiagonal line.

In matrix operations, the sum of antidiagonal elements can be a crucial calculation.

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#### Diagonal

A matrix where all elements outside the main diagonal are zero.

In a diagonal matrix, the elements along the diagonal line, like (1,1) or (2,2), are the only non-zero values.

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#### Antidiagonal

A line connecting opposite corners of a square matrix, from top-right to bottom-left.

In a 3x3 matrix, the antidiagonal includes the elements at positions (1,3), (2,2), and (3,1).

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#### Diagonal

A pattern or shape that is symmetrical along a diagonal line.

The pattern on the tile showed perfect diagonal symmetry, forming a harmonious design.

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#### Antidiagonal

In geometry, a line that extends from the top-right to the bottom-left of a shape.

The antidiagonal line divided the square into two equal triangles.

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#### Diagonal

A line segment that joins two non-adjacent vertices in a polygon.

The geometric design was made of several diagonal lines crossing each other.

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#### Antidiagonal

A type of symmetry in patterns or designs, mirrored along an antidiagonal line.

The quilt pattern showed antidiagonal symmetry, with colors mirroring across the top-right to bottom-left axis.

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#### Antidiagonal

A square matrix where all elements are zero except those on the antidiagonal.

In an antidiagonal matrix, the entry at (1,3) might be 5 in a 3x3 matrix, with other non-antidiagonal elements being zero.

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#### Antidiagonal

(linear algebra) The diagonal of a matrix that leads from top-right towards bottom-left.

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#### Diagonal

Relating to or being the front left and back right feet or the front right and back left feet of a quadruped.

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#### Diagonal

Something, such as a row, course, or part, that is arranged obliquely.

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#### Diagonal

(geometry) Joining two nonadjacent vertices (of a polygon or polyhedron).

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#### Diagonal

Of or related to the cater-corner (diagonally opposite) legs of a quadruped, whether the front left and back right or front right and back left.

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#### Diagonal

Joining two not adjacent angles of a quadrilateral or multilateral figure; running across from corner to corner; crossing at an angle with one of the sides.

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#### Diagonal

A right line drawn from one angle to another not adjacent, of a figure of four or more sides, and dividing it into two parts.

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#### Diagonal

A member, in a framed structure, running obliquely across a panel.

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#### Diagonal

A diagonal cloth; a kind of cloth having diagonal stripes, ridges, or welts made in the weaving.

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#### Diagonal

(geometry) a straight line connecting any two vertices of a polygon that are not adjacent

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#### Diagonal

A line or cut across a fabric that is not at right angles to a side of the fabric

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#### Diagonal

An oblique line of squares of the same color on a checkerboard;

The bishop moves on the diagonals

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#### Diagonal

(mathematics) a set of entries in a square matrix running diagonally either from the upper left to lower right entry or running from the upper right to lower left entry

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#### Diagonal

A punctuation mark (/) used to separate related items of information

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#### Diagonal

At an angle; especially connecting two nonadjacent corners of a plane figure or any two corners of a solid that are not in the same face;

A diagonal line across the page

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#### Diagonal

A straight line connecting opposite corners of a polygon, especially a square or rectangle.

The diagonal of the rectangle was longer than any of its sides.

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#### Diagonal

Cutting or dividing something along a diagonal line.

She preferred her sandwich cut diagonally, creating two triangles.

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## Repeatedly Asked Queries

#### What is diagonal symmetry?

Symmetry along a diagonal line, often seen in patterns and designs.

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#### How does a diagonal differ from an antidiagonal?

A diagonal connects the top-left to the bottom-right, while an antidiagonal does the opposite.

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#### What is a diagonal matrix?

A matrix where all elements outside the main diagonal are zero.

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#### What is an antidiagonal?

It's a line in a square matrix connecting the top-right corner to the bottom-left corner.

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#### Can rectangles have diagonals?

Yes, any quadrilateral, including rectangles, has diagonals.

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#### How do diagonals affect the properties of polygons?

They can help in calculating area and understanding symmetry.

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#### Can antidiagonals exist outside of square matrices?

Conceptually, yes, but they're primarily defined in square matrices.

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#### Can a matrix have both diagonal and antidiagonal elements?

Yes, but they will be on different lines unless it's a single element in a 1x1 matrix.

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#### Are diagonal lines always straight?

In geometry, yes, diagonals are straight lines.

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#### How are antidiagonals used in mathematics?

They are used in matrix operations and to denote a specific type of symmetry.

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#### Do diagonals always bisect each other?

In certain shapes like squares and rectangles, yes.

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#### Do diagonal cuts change the shape of an object?

Yes, they can transform shapes, like turning a square into two triangles.

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#### Is there a specific use for diagonal lines in art?

Yes, they often create a sense of movement or perspective in artwork.

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#### How are antidiagonal elements identified in a matrix?

By the condition that the row number plus column number equals the size of the matrix plus one.

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#### What is an antidiagonal matrix?

A square matrix with all non-antidiagonal elements being zero.

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#### Can a diagonal be a line of symmetry?

Yes, especially in squares and equilateral triangles.

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#### Are diagonals equal in all squares?

Yes, in a square, all diagonals are of equal length.

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#### Is every square matrix diagonal?

No, only if all non-diagonal elements are zero.

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#### Are all diagonal matrices invertible?

Not necessarily; it depends on the elements on the diagonal.

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#### Can antidiagonal elements be the same in a matrix?

Yes, they can be, but it's not a requirement.

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About Author

Written by

Shumaila SaeedShumaila Saeed, an expert content creator with 6 years of experience, specializes in distilling complex topics into easily digestible comparisons, shining a light on the nuances that both inform and educate readers with clarity and accuracy.