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2 results

  • 14 - Normal Matrices and the Spectral Theorem

  • Stephan Ramon Garcia, Pomona College, California, Roger A. Horn, University of Utah
  • Book:
    Matrix Mathematics
    Published online:
    29 September 2023
    Print publication:
    25 May 2023, pp 277-300

  • Summary

    Chapter 14: Every square complex matrix is unitarily similar to an upper triangular matrix, but which matrices are unitarily similar to a diagonal matrix? The answer is the main result of this chapter: the spectral theorem for normal matrices. Hermitian, skew-Hermitian, unitary, and circulant matrices are unitarily diagonalizable. As a consequence, they have special properties that we investigate in this and following chapters.

  • 2 - Algebraic Eigenproblems and Their Applications

  • from Part I - Matrix Methods
  • Kevin W. Cassel, Illinois Institute of Technology
  • Book:
    Matrix, Numerical, and Optimization Methods in Science and Engineering
    Published online:
    18 February 2021
    Print publication:
    04 March 2021, pp 64-154

  • Summary

    The algebraic eigenproblem is the mathematical answer to the physical questions:What are the principal stresses in a solid or fluid and on what planes do they act?What are the natural frequencies of a system?Is the system stable to small disturbances?What is the best basis with respect to which to solve a system of linear algebraic equations with a real symmetric coefficient matrix?What is the best basis with respect to which to solve a system of linear ordinary differential equations?What is the best basis with respect to which to represent an experimental or numerical data set?