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main diagonal

C1
UK/ˌmeɪn daɪˈæɡənəl/US/ˌmeɪn daɪˈæɡənəl/

Formal / Technical

My Flashcards

Definition

Meaning

The diagonal of a square matrix from the top-left to bottom-right corner, containing entries where the row and column indices are equal.

The set of elements a_ii (i = 1, 2, ..., n) of an n-by-n matrix; in geometry, it can also refer to the primary diagonal connecting opposite vertices of a polygon or polyhedron, especially in a square or cube.

Linguistics

Semantic Notes

Always used in contexts involving matrices (linear algebra), geometry, or structured data arrays. The concept is abstract and strictly defined. For non-square matrices, the term 'main diagonal' is sometimes used but more precisely called the 'leading diagonal'.

Dialectal Variation

British vs American Usage

Differences

No significant lexical or definitional differences. In British mathematical contexts, 'leading diagonal' is a more common alternative term.

Connotations

Purely technical, no cultural connotations.

Frequency

Slightly higher frequency in academic and engineering contexts in both regions. The American term 'main diagonal' is dominant globally in computational sciences.

Vocabulary

Collocations

strong
trace of the main diagonalelements on the main diagonalentries along the main diagonal
medium
identity matrix main diagonalsum of the main diagonalzero off the main diagonal
weak
main diagonal entriesmain diagonal matrixmain diagonal of a table

Grammar

Valency Patterns

The main diagonal of [MATRIX]Elements/Lies on the main diagonalSum/Trace along the main diagonal

Vocabulary

Synonyms

Strong

diagonal (in context)major diagonal

Neutral

principal diagonalleading diagonalprimary diagonal

Weak

central diagonalkey diagonal

Vocabulary

Antonyms

off-diagonalanti-diagonalsecondary diagonal

Phrases

Idioms & Phrases

  • “None”

Usage

Context Usage

Business

Rare, except in data analysis or financial modelling involving correlation matrices.

Academic

Very common in mathematics, physics, computer science, engineering, and statistics textbooks and papers.

Everyday

Extremely rare; only in specific technical discussions.

Technical

The primary context. Used in linear algebra, matrix computations, scientific programming (e.g., NumPy's `numpy.diag`), and geometry.

Examples

By Part of Speech

verb

British English

  • The function `diag` can main-diagonalise the vector.
  • To solve, we must main-diagonalise the operator (rare).

American English

  • The algorithm will main-diagonalize the extracted values.
  • We need to main-diagonalize the transformation matrix (rare).

adverb

British English

  • The values are arranged main-diagonally.
  • The data populated the table main-diagonally (highly rare).

American English

  • The numbers are placed main-diagonally in the array.
  • The function reads the data main-diagonally (highly rare).

adjective

British English

  • The main-diagonal elements are all positive.
  • Observe the main-diagonal dominance of the matrix.

American English

  • The main-diagonal entries were highlighted.
  • Check for main-diagonal dominance in the system.

Examples

By CEFR Level

A2
  • The picture has a line from one corner to the opposite corner; this is a diagonal line. (Concept introduction)
B1
  • In a simple data table, sometimes the most important information is found on the diagonal from the top-left.
  • A square has two diagonals, but in some maths problems we focus on one.
B2
  • For a 3x3 identity matrix, the main diagonal consists of ones and all other entries are zero.
  • The sum of the numbers on the main diagonal of the square grid gave us the first result.
C1
  • The stability of the system depends on the eigenvalues, which are intrinsically linked to the trace of the main diagonal.
  • The programmer used `numpy.diag()` to extract the main diagonal of the covariance matrix for further analysis.

Learning

Memory Aids

Mnemonic

Imagine a square city grid. The 'MAIN street' runs from the northwest gate (top-left) directly to the southeast gate (bottom-right). This is the MAIN route, just like the MAIN DIAGONAL of a matrix.

Conceptual Metaphor

THE BACKBONE OF A STRUCTURE (The main diagonal provides critical structural information about a matrix, much like a backbone defines an organism's core structure.)

Watch out

Common Pitfalls

Translation Traps (for Russian speakers)

  • Do not literally translate as 'главная диагональ' without confirming context, though it is often correct in mathematics. In non-mathematical geometry, 'principal diagonal' might be used.
  • Avoid confusing with 'диагональ' alone, which can mean any diagonal line or a television screen.

Common Mistakes

  • Using 'main diagonal' for non-square matrices.
  • Confusing 'main diagonal' with the diagonal from top-right to bottom-left (the anti-diagonal).
  • Misspelling as 'main diagnol'.
  • Using in non-technical contexts where 'diagonal line' is sufficient.

Practice

Quiz

Fill in the gap
In linear algebra, the sum of the elements on the of a square matrix is called its trace.
Multiple Choice

In which of these matrices do all non-zero elements lie strictly on the main diagonal?

FAQ

Frequently Asked Questions

Strictly speaking, yes. For rectangular matrices, the concept of a diagonal from top-left to bottom-right is not formally defined, though some programming libraries may refer to it for an M-by-N matrix where M ≠ N.

The main diagonal runs from top-left (a₁₁) to bottom-right (aₙₙ). The anti-diagonal (or counter-diagonal) runs from top-right (a₁ₙ) to bottom-left (aₙ₁).

Yes. The entries on the main diagonal can be any numbers, including zeros. A matrix where all non-diagonal elements are zero is called a diagonal matrix, even if some diagonal entries are zero.

It is crucial for defining key matrix properties: the trace (sum of diagonal elements), the determinant (influenced by diagonal values), and for identifying special matrices like diagonal, identity, or triangular matrices.