Introduction to Matrix Theory矩阵理论引论

Chapter 1  Introduction to Linear Algebra1.1  The linear space1.1.1  Fields and mappings1.1.2  Definition of the linear space1.1.3  Basis and dimension1.1.4  Coordinate1.1.5  Transformations of bases and coordinates1.1.6  Subspace and the dimension theorem for vector spaces1.2  Linear transformation and matrices1.2.1  Linear transformation1.2.2  Matrices of linear transformations and isomorphism1.3  Eigenvalues and the Jordan canonical form1.3.1  Eigenvalues and eigenvectors1.3.2  Diagonal matrices1.3.3  Schur's theorem and the Cayley- Hamilton theorem1.3.4  The Jordan canonical form1.4  Unitary spacesExercise 1Chapter 2  Matrix Analysis2.1  Vector norm2.2  Matrix norm2.3  Matrix sequences and series2.4  Matrix function2.5  Differentiation and integration of matrices2.6  Applications of matrix functions2.7  Estimation of eigenvaluesExercise 2Chapter 3  Matrix Decomposition3.1  QR decomposition3.2  Full rank decomposition3.3  Singular value decomposition3.4  The spectral decompositionExercise 3Chapter 4  Generalized Inverse4.1  The generalized inverse of a matrix4.2 A{1},A{1,3} andA{1,4}4.3  The Moore- Penrose inverse A+4.4  The generalized inverses and the linear equationsExercise 4Chapter 5  Tensor Product5.1  Definition and properties of the tensor product5.2  The tensor product and eigenvalues5.3  Straighten operation on matrices5.4  The tensor product and matrix equationExercise 5Chapter 6  Introduction To Nonnegative Matrices6.1  Preliminary properties on nonnegative matrices6.2  Positive matrices and the Perron theorem6.3  Irreducible nonnegative matrices6.4  Primitive matrices and M matrices6.5  Stochastic matrices6.6  Two models of nonnegative matricesExercise 6References