README.md

Chapter 0 Preliminaries

This chapter is mainly about the preliminary knowledge required by the main text. If you are familiar with QR, eigen and Schur decomposition of matrices (especially tridiagonal matrices and Hessenberg matrices), or you do not care about these algorithms which have little to do with Krylov subspace methods, you can only read this chapter's first section.

• Chapter 0 Preliminaries (Page 1)
  • Section 1 Notations (Page 1)
    • 1.1 Scalars, Vectors and Matrices (Page 1)
    • 1.2 Inner Product and Norm (Page 2)
    • 1.3 Eigen Decomposition (Page 2)
    • 1.4 Iteration Notation (Page 3)
  • Section 2 QR Factorization (Page 3)
    • 2.1 Definition (Page 3)
    • 2.2 Householder QR Factorization (Page 4)
    • 2.3 QR Factorization of Special Matrices (Page 6)
  • Section 3 Eigen Decomposition of Symmetric Tridiagonal Matrices (Page 6)
  • Section 4 Schur Decomposition (Page 12)
    • 4.1 Definition (Page 12)
    • 4.2 Schur Decomposition of Hessenberg Matrices (Page 13)
    • 4.3 Reorder of Schur Decomposition (Page 17)
  • Section 5 Eigen Decomposition of Hessenberg Matrices (Page 19)
  • Section 6 Other Proofs (Page 23)
    • 6.1 Rayleigh Quotient (Page 23)
    • 6.2 Multiplicity of Real Symmetric Matrices (Page 24)

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