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2 results

  • 3 - Matrix Factorization: Eigendecomposition and SVD

  • Jeffrey A. Fessler, University of Michigan, Ann Arbor, Raj Rao Nadakuditi, University of Michigan, Ann Arbor
  • Book:
    Linear Algebra for Data Science, Machine Learning, and Signal Processing
    Published online:
    01 November 2024
    Print publication:
    16 May 2024, pp 63-95

  • Summary

    This chapter introduces matrix factorizations – somewhat like the reverse of matrix multiplication. It starts with the eigendecomposition of symmetric matrices, then generalizes to normal and asymmetric matrices. It introduces the basics of the singular value decomposition (SVD) of general matrices. It discusses a simple application of the SVD that uses the largest singular value of a matrix (the spectral norm), posed as an optimization problem, and then describes optimization problems related to eigenvalues and the smallest singular value. (The “real” SVD applications appear in subsequent chapters.) It discusses the special situations when one can relate the eigendecomposition and an SVD of a matrix, leading to the special class of positive (semi)definite matrices. Along the way there are quite a few small eigendecomposition and SVD examples.

  • The Spectral Radius of a Non-Negative Matrix

  • A. Berman
  • Journal:
    Canadian Mathematical Bulletin / Volume 21 / Issue 1 / 01 March 1978
    Published online by Cambridge University Press:
    20 November 2018, pp. 113-114
    Print publication:
    01 March 1978
    • Article
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  • A max min formula for the spectral radius of a non-negative matrix is derived from a characterization of non-singular M-matrices in terms of diagonal stability.