希望对大家有用，免费分享了
Preface v
1 Matrices 1
1.1 Basic Terminology . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Basic Operations . . . . . . . . . . . . . . . . . . . . . . . . 2
1.3 Some Basic Types of Matrices . . . . . . . . . . . . . . . . . 6
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2 Submatrices and Partitioned Matrices 13
2.1 Some Terminology and Basic Results . . . . . . . . . . . . . . 13
2.2 Scalar Multiples, Transposes, Sums, and Products of Partitioned
Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.3 Some Results on the Product of a Matrix and a Column Vector . 19
2.4 Expansion of a Matrix in Terms of Its Rows, Columns, or Elements
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
3 Linear Dependence and Independence 23
3.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
3.2 Some Basic Results . . . . . . . . . . . . . . . . . . . . . . . 24
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
4 Linear Spaces: Row and Column Spaces 27
4.1 Some Definitions, Notation, and Basic Relationships and Properties
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
xii
4.2 Subspaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
4.3 Bases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
4.4 Rank of a Matrix . . . . . . . . . . . . . . . . . . . . . . . . 36
4.5 Some Basic Results on Partitioned Matrices and on Sums of Matrices
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
5 Trace of a (Square) Matrix 49
5.1 Definition and Basic Properties . . . . . . . . . . . . . . . . . 49
5.2 Trace of a Product . . . . . . . . . . . . . . . . . . . . . . . 50
5.3 Some Equivalent Conditions . . . . . . . . . . . . . . . . . . 52
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
6 Geometrical Considerations 55
6.1 Definitions: Norm, Distance, Angle, Inner Product, and Orthogonality
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
6.2 Orthogonal and Orthonormal Sets . . . . . . . . . . . . . . . 61
6.3 Schwarz Inequality . . . . . . . . . . . . . . . . . . . . . . . 62
6.4 Orthonormal Bases . . . . . . . . . . . . . . . . . . . . . . . 63
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
7 Linear Systems: Consistency and Compatibility 71
7.1 Some Basic Terminology . . . . . . . . . . . . . . . . . . . . 71
7.2 Consistency . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
7.3 Compatibility . . . . . . . . . . . . . . . . . . . . . . . . . . 73
7.4 Linear Systems of the Form A
AX  A
B . . . . . . . . . . . 74
Exercise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
8 Inverse Matrices 79
。。。。。。