secondary diagonal

Thân thiện

A student draws the secondary diagonal on a whiteboard matrix.

Definition

Noun: - In a square matrix, the diagonal that runs from the bottom-left element to the top-right element. It is the set of entries where the row and column indices sum to a constant value (typically n+1 for an n×n matrix, where indices start at 1).

Usage

The term is used specifically in the context of linear algebra and matrix theory to distinguish this diagonal from the main (or primary) diagonal, which runs from the top-left to the bottom-right. - In a 3x3 matrix, the secondary diagonal consists of the elements at positions (3,1), (2,2), and (1,3) if using 1-based indexing. - The sum of the elements on the secondary diagonal is sometimes called the anti-diagonal sum.

Advanced Usage

• Anti-diagonal: A common synonym for the secondary diagonal.
• Counter-diagonal: Another less common synonym.
• The secondary diagonal is significant in operations such as finding the trace of the anti-diagonal or checking for matrix properties like persymmetry (symmetry across the secondary diagonal).

Variants and Related Words

• Main diagonal / Primary diagonal: The diagonal from the top-left to the bottom-right of a square matrix.
• Anti-diagonal matrix: A square matrix where all entries are zero except those on the secondary diagonal.
• Persymmetric matrix: A matrix that is symmetric with respect to its secondary diagonal.

Synonyms

• Anti-diagonal
• Counter-diagonal
• Minor diagonal (Note: This term is less common and can be ambiguous, as it sometimes refers to the diagonal of a minor in a matrix.)

Related Phrases and Concepts

• Trace of the anti-diagonal: The sum of the elements on the secondary diagonal.
• Flip across the secondary diagonal: A matrix operation reflecting elements over this line.

Học thuật

Noun

1. the diagonal of a square matrix running from the lower left entry to the upper right entry