Definition

1. Noun:
   • A square matrix with all elements not on the main diagonal equal to zero: A diagonal matrix is a specific type of square matrix where the only non-zero entries are located on the main diagonal, which runs from the top-left to the bottom-right corner. All other entries (off-diagonal entries) are zero.

Usage

• Noun:
  • The identity matrix is a special case of a diagonal matrix where all diagonal entries are 1.
  • In linear algebra, multiplying by a diagonal matrix scales the rows or columns of another matrix.
  • The eigenvalues of a diagonal matrix are simply the entries on its main diagonal.

Examples

• Noun:
  • The matrix [[2, 0, 0], [0, -1, 0], [0, 0, 5]] is a diagonal matrix.
  • Solving systems of equations is simpler when the coefficient matrix is a diagonal matrix.
  • A scalar matrix, where all diagonal elements are identical, is a type of diagonal matrix.

Advanced Usage

• "To be diagonalizable": A matrix is diagonalizable if it is similar to a diagonal matrix. This means it can be written in the form , where is a diagonal matrix.
  • A symmetric matrix is always diagonalizable by an orthogonal matrix.

Variants and Related Words

• Diagonal (adj/n): Relating to a straight line joining two opposite corners; in a matrix context, it refers to the main diagonal.
  • The trace of a matrix is the sum of its diagonal elements.
• Scalar Matrix (n): A diagonal matrix where all the diagonal entries are equal.
  • The identity matrix is a scalar matrix with all entries equal to 1.
• Tridiagonal Matrix (n): A matrix that has non-zero elements only on the main diagonal and the diagonals directly above and below it. This is a related but distinct concept.
  • Many numerical algorithms are efficient for tridiagonal matrices.

Synonyms

• No direct single-word synonyms in common mathematical parlance. The term is precise. It can be described as:
  • A matrix with off-diagonal zeros.

Related Phrases

• Main diagonal / Principal diagonal: The diagonal of a square matrix from the top-left to the bottom-right entry.
  • The values on the main diagonal define a diagonal matrix.
• Off-diagonal elements: The elements of a matrix that are not on the main diagonal.
  • In a diagonal matrix, all off-diagonal elements are zero.

Noun

1. a square matrix with all elements not on the main diagonal equal to zero