Chapter 1  Introduction to Linear Algebra

  1.1  The linear space

    1.1.1  Fields and mappings

    1.1.2  Definition of the linear space

    1.1.3  Basis and dimension

    1.1.4  Coordinate

    1.1.5  Transformations of bases and coordinates

    1.1.6  Subspace and the dimension theorem for vector spaces

  1.2  Linear transformation and matrices

    1.2.1  Linear transformation

    1.2.2  Matrices of linear transformations and isomorphism

  1.3  Eigenvalues and the Jordan canonical form

    1.3.1  Eigenvalues and eigenvectors

    1.3.2  Diagonal matrices

    1.3.3  Schur's theorem and the Cayley- Hamilton theorem

    1.3.4  The Jordan canonical form

  1.4  Unitary spaces

  Exercise 1

Chapter 2  Matrix Analysis

  2.1  Vector norm

  2.2  Matrix norm

  2.3  Matrix sequences and series

  2.4  Matrix function

  2.5  Differentiation and integration of matrices

  2.6  Applications of matrix functions

  2.7  Estimation of eigenvalues

  Exercise 2

Chapter 3  Matrix Decomposition

  3.1  QR decomposition

  3.2  Full rank decomposition

  3.3  Singular value decomposition

  3.4  The spectral decomposition

  Exercise 3

Chapter 4  Generalized Inverse

  4.1  The generalized inverse of a matrix

  4.2 A{1},A{1,3} andA{1,4}

  4.3  The Moore- Penrose inverse A+

  4.4  The generalized inverses and the linear equations

  Exercise 4

Chapter 5  Tensor Product

  5.1  Definition and properties of the tensor product

  5.2  The tensor product and eigenvalues

  5.3  Straighten operation on matrices

  5.4  The tensor product and matrix equation

  Exercise 5

Chapter 6  Introduction To Nonnegative Matrices

  6.1  Preliminary properties on nonnegative matrices

  6.2  Positive matrices and the Perron theorem

  6.3  Irreducible nonnegative matrices

  6.4  Primitive matrices and M matrices

  6.5  Stochastic matrices

  6.6  Two models of nonnegative matrices

  Exercise 6

References