Definition

1. Noun:
   • A matrix with the same number of rows and columns: A "square matrix" is a mathematical array of numbers, symbols, or expressions arranged in rows and columns, where the number of rows is equal to the number of columns. This property defines its fundamental structure and distinguishes it from rectangular matrices.

Usage Examples

• Noun:
  • The identity matrix is a fundamental example of a square matrix.
  • To calculate the determinant, you must first have a square matrix.
  • The problem requires you to invert a 3x3 square matrix.

Advanced Usage

• "Principal diagonal of a square matrix": Refers to the diagonal line of elements from the top-left to the bottom-right corner.

  • The trace of a square matrix is the sum of the elements on its principal diagonal.

• "Singular square matrix": A square matrix that does not have a multiplicative inverse, typically because its determinant is zero.

  • If the determinant is zero, the square matrix is singular and cannot be inverted.

Variants and Related Words

• Matrix (n): A rectangular array of numbers or other mathematical objects.

  • Data is often organized in a matrix for computational analysis.

• Determinant (n): A scalar value that can be computed from the elements of a square matrix.

  • The determinant provides important information about the properties of the square matrix.

• Eigenvalue (n): A special set of scalars associated with a linear transformation represented by a square matrix.

  • Finding the eigenvalues is a key step in analyzing a square matrix.

Synonyms

• n-by-n matrix: A technical synonym emphasizing the equal dimensions (n rows and n columns).

Related Phrases

• "Invert a square matrix": To find another matrix which, when multiplied by the original, yields the identity matrix.

  • Not all square matrices can be inverted; only those with a non-zero determinant.

• "Diagonalize a square matrix": To express a square matrix in a form where all non-diagonal elements are zero, if possible.

  • The goal is to diagonalize the square matrix to simplify calculations.

Noun

1. a matrix with the same number of rows and columns