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2 results

  • LEFT AND RIGHT EIGENVECTORS OF A VARIANT OF THE SYLVESTER–KAC MATRIX

  • Part of
  • WENCHANG CHU, EMRAH KILIÇ
  • Journal:
    Bulletin of the Australian Mathematical Society / Volume 109 / Issue 2 / April 2024
    Published online by Cambridge University Press:
    19 June 2023, pp. 316-326
    Print publication:
    April 2024
    • Article
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  • As an extension of Sylvester’s matrix, a tridiagonal matrix is investigated by determining both left and right eigenvectors. Orthogonality relations between left and right eigenvectors are derived. Two determinants of the matrices constructed by the left and right eigenvectors are evaluated in closed form.

  • 7 - Unitary Matrices

  • 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 137-157

  • Summary

    Chapter 7: Unitary matrices play important roles in theory and computation. The adjoint of a unitary matrix is its inverse, so unitary matrices are easy to invert. They preserve lengths and angles, and have remarkable stability properties in many numerical algorithms. In this chapter, we explore the properties of unitary matrices and present several special cases. We derive an explicit formula for a unitary matrix whose first column is given. We give a constructive proof of the QR factorization and show that every square complex matrix is unitarily similar to an upper Hessenberg matrix.