Definition

1. Noun:
   • The diagonal of a square matrix running from the upper left entry to the lower right entry: In mathematics, specifically in linear algebra, the "principal diagonal" refers to the sequence of elements in a square matrix that extend from the top-left corner to the bottom-right corner.

Usage

• The term is used exclusively in the context of matrix algebra and related mathematical fields.
• It is a specific, technical term. When referring to this concept, the phrase "principal diagonal" is standard.

Examples

• Noun:
  • The sum of the elements on the principal diagonal is called the trace of the matrix.
  • In an identity matrix, all entries on the principal diagonal are ones, and all other entries are zeros.
  • The eigenvalues of a triangular matrix are precisely the entries on its principal diagonal.

Advanced Usage

• "to lie on the principal diagonal": Describes the position of an element within the matrix.
  • The element a₁₁ lies on the principal diagonal.
• The concept is fundamental for defining other matrix properties, such as being a diagonal matrix (where all non-principal-diagonal elements are zero) or a trace (the sum of the principal diagonal elements).

Variants and Related Words

• Main diagonal: A completely synonymous term for "principal diagonal."
• Leading diagonal: Another synonymous term, less common but acceptable.
• Diagonal (noun): A more general term. In the context of a square matrix, "the diagonal" often implicitly means the principal diagonal, but it can also refer to other diagonals (like the anti-diagonal).
• Anti-diagonal (noun): The diagonal running from the upper right entry to the lower left entry, which is contrasted with the principal diagonal.

Synonyms

• Main diagonal
• Leading diagonal

Related Phrases

• Off-diagonal: Refers to all elements of a square matrix that are on the principal diagonal.
  • The covariance was only present in the off-diagonal elements.

Noun

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