Definition

1. Noun:
   • Characteristic root of a square matrix: In mathematics, this is a number (often denoted by λ) such that when it is multiplied by the identity matrix and subtracted from a given square matrix, the resulting matrix has a determinant of zero. It is a scalar value associated with a linear transformation represented by the matrix.

Usage

• This term is used exclusively in the field of linear algebra. It describes a fundamental scalar associated with a matrix. Finding these roots is central to analyzing the matrix's properties, such as its stability, diagonalizability, and the behavior of systems it represents.
• Formulaic Context: If A is an square matrix, then λ is a characteristic root of a square matrix A if , where I is the identity matrix and denotes the determinant.

Examples

• Noun:
  • To understand the system's long-term behavior, the engineer calculated the characteristic root of the square matrix representing its dynamics.
  • The stability of the solution depends on whether the real parts of all characteristic roots of the square matrix are negative.

Advanced Usage

• "To find the characteristic roots of a square matrix": This is a standard phrase describing the process of solving the characteristic equation for λ.
  • The first step in diagonalizing a matrix is to find all its characteristic roots.
• "Dominant characteristic root": Refers to the eigenvalue with the largest absolute value, which often governs the growth rate in iterative processes.
  • The population model's growth rate is determined by the dominant characteristic root of its transition matrix.

Variants and Related Words

• Eigenvalue (n): A completely synonymous term. "Characteristic root" and "eigenvalue" are used interchangeably in linear algebra.
  • λ is an eigenvalue of matrix A.
• Characteristic value (n): Another less common synonym.
• Latent root (n): A synonym occasionally used in some statistical contexts.
• Spectrum (n): The set of all characteristic roots of a matrix.
  • The spectrum of the operator was analyzed.

Synonyms

• Eigenvalue: The most common synonym.
• Proper value: An older term, now less frequent.
• Latent root: Used in specific fields like factor analysis.

Related Phrases

• Characteristic equation: The equation whose solutions are the characteristic roots.
  • Solving the characteristic equation yielded two complex roots.
• Characteristic polynomial: The polynomial in λ obtained by expanding .
  • The degree of the characteristic polynomial equals the size of the matrix.

Notes

• The term is inherently technical. In most modern contexts, especially in physics and engineering, the word "eigenvalue" is more prevalent. "Characteristic root" is equally correct but may be seen more in pure mathematical or older texts.
• It is always associated with a matrix; non-square matrices do not have characteristic roots.

Noun

1. (mathematics) any number such that a given square matrix minus that number times the identity matrix has a zero determinant