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| Introduction | |
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| The scope of matrix algebra | |
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| General description of a matrix | |
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| Subscript notation | |
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| Summation notation | |
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| Dot notation | |
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| Definition of a matrix | |
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| Vectors and scalars | |
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| General notation | |
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| Illustrative examples | |
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| Exercises | |
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| Basic Operations | |
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| The transpose of a matrix | |
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| A reflexive operation | |
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| Vectors | |
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| Partitioned matrices | |
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| Example | |
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| General specification | |
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| Transposing a partitioned matrix | |
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| Partitioning into vectors | |
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| The trace of a matrix | |
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| Addition | |
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| Scalar multiplication | |
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| Subtraction | |
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| Equality and the null matrix | |
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| Multiplication | |
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| The inner product of two vectors | |
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| A matrix-vector product | |
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| A product of two matrices | |
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| Existence of matrix products | |
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| Products with vectors | |
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| Products with scalars | |
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| Products with null matrices | |
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| Products with diagonal matrices | |
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| Identity matrices | |
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| The transpose of a product | |
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| The trace of a product | |
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| Powers of a matrix | |
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| Partitioned matrices | |
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| Hadamard products | |
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| The Laws of algebra | |
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| Associative laws | |
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| The distributive law | |
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| Commutative laws | |
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| Contrasts with scalar algebra | |
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| Exercises | |
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| Special Matrices | |
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| Symmetric matrices | |
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| Products of symmetric matrices | |
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| Properties of AA' and A'A | |
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| Products of vectors | |
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| Sums of outer products | |
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| Elementary vectors | |
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| Skew-symmetric matrices | |
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| Matrices having all elements equal | |
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| Idempotent matrices | |
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| Orthogonal matrices | |
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| Definitions | |
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| Special cases | |
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| Quadratic forms | |
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| Positive definite matrices | |
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| Exercises | |
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| Determinants | |
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| Expansion by minors | |
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| First- and second-order determinants | |
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| Third-order determinants | |
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| n-order determinants | |
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| Formal definition | |
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| Basic properties | |
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| Determinant of a transpose | |
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| Two rows the same | |
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| Cofactors | |
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| Adding multiples of a row (column) to a row (column) | |
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| Products | |
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| Elementary row operations | |
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| Factorization | |
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| A row (column) of zeros | |
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| Interchanging rows (columns) | |
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| Adding a row to a multiple of a row | |
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| Examples | |
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| Diagonal expansion | |
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| The Laplace expansion | |
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| Sums and differences of determinants | |
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| Exercises | |
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| Inverse Matrices | |
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| Introduction: solving equations | |
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| Products equal to I | |
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| Cofactors of a determinant | |
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| Derivation of the inverse | |
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| Conditions for existence of the inverse | |
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| Properties of the inverse | |
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| Some simple special cases | |
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| Inverses of order 2 | |
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| Diagonal matrices | |
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| I and J matrices | |
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| Orthogonal matrices | |
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| Idempotent matrices | |
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| Equations and algebra | |
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| Solving linear equations | |
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| Algebraic simplifications | |
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| Computers and inverses | |
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| The arithmetic of linear equations | |
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| Rounding error | |
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| Left and right inverses | |
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| Exercises | |
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| Rank | |
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| Linear combinations of vectors | |
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| Linear transformations | |
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| Linear dependence and independence | |
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| Definitions | |
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| General characteristics | |
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| Linearly dependent vectors | |
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| At least two a's are nonzero | |
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| Vectors are linear combinations of others | |
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| Partitioning matrices | |
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| Zero determinants | |
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| Inverse matrices | |
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| Testing for dependence (simple cases) | |
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| Linearly independent (LIN) vectors | |
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| Nonzero determinants and inverse matrices | |
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| Linear combinations of LIN vectors | |
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| A maximum number of LIN vectors | |
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| The number of LIN rows and columns in a matrix | |
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| The rank of a matrix | |
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| Rank and inverse matrices | |
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| Permutation matrices | |
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| Full-rank factorization | |
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| Basic development | |
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| The general case | |
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| Matrices of full row (column) rank | |
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| Vector spaces | |
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| Euclidean space | |
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| Vector spaces | |
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| Spanning sets and bases | |
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| Many spaces of order n | |
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| Subspaces | |
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| The range and null space of a matrix | |
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| Exercises | |
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| Canonical Forms | |
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| Elementary operators | |
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| Row operations | |
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| Transposes | |
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| Column operations | |
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| Inverses | |
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| Rank and the elementary operators | |
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| Rank | |
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| Products of elementary operators | |
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| Equivalence | |
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| Finding the rank of a matrix | |
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| Some special LIN vectors | |
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| Calculating rank | |
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| A general procedure | |
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| Reduction to equivalent canonical form | |
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| Row operations | |
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| Column operations | |
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| The equivalent canonical form | |
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| Non-uniqueness of P and Q | |
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| Existence is assured | |
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| Full-rank factorization | |
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| Rank of a product matrix | |
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| Symmetric matrices | |
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| Row and column operations | |
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| The diagonal form | |
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| The canonical form under congruence | |
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| Two special provisions | |
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| Full-rank factorization | |
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| Non-negative definite matrices | |
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| Diagonal elements and principal minors | |
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| Congruent canonical form | |
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| Full-rank factorization | |
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| Quadratic forms as sums of squares | |
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| Full row (column) rank matrices | |
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| Exercises | |
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| Generalized Inverses | |
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| The Moore-Penrose inverse | |
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| Generalized inverses | |
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| Derivation from row operations | |
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| Derivation from the diagonal form | |
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| Other names and symbols | |
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| An algorithm | |
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| An easy form | |
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| A general form | |
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| Arbitrariness in a generalized inverse | |
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| Symmetric matrices | |
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| Non-negative definite matrices | |
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| A general algorithm | |
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| The matrix X'X | |
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| Exercises | |
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| Solving Linear Equations | |
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| Equations having many solutions | |
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| Consistent equations | |
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| Definition | |
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| Existence of solutions | |
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| Tests for consistency | |
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| Equations having one solution | |
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| Deriving solutions using generalized inverses | |
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| Obtaining a solution | |
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| Obtaining many solutions | |
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| All possible solutions | |
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| Combinations of solutions | |
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| Linearly independent solutions | |
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| An invariance property | |
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| Equations Ax = 0 | |
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| General properties | |
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| Orthogonal solutions | |
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| Orthogonal vector spaces | |
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| A complete example | |
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| Least squares equations | |
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| Exercises | |
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| Partitioned Matrices | |
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| Orthogonal matrices | |
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| Determinants | |
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| Inverses | |
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| Schur complements | |
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| Generalized inverses | |
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| Direct sums | |
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| Direct products | |
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| Exercises | |
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| Eigenvalues and Eigenvectors | |
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| Introduction: age distribution vectors | |
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| Derivation of eigenvalues | |
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| Elementary properties of eigenvalues | |
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| Eigenvalues of powers of a matrix | |
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| Eigenvalues of a scalar-by-matrix product | |
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| Eigenvalues of polynomials | |
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| The sum and product of eigenvalues | |
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| Calculating eigenvectors | |
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| A general method | |
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| Simple roots | |
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| Multiple roots | |
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| The similar canonical form | |
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| Derivation | |
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| Uses | |
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| Symmetric matrices | |
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| Eigenvalues all real | |
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| Symmetric matrices are diagonable | |
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| Eigenvectors are orthogonal | |
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| Rank equals number of nonzero eigenvalues | |
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| Dominant eigenvalues | |
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| Factoring the characteristic equation | |
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| Exercises | |
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| Appendix to Chapter 11 | |
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| Proving the diagonability theorem | |
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| The number of nonzero eigenvalues never exceeds rank | |
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| A lower bound on r(A - [lambda subscript k]I) | |
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| Proof of the diagonability theorem | |
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| All symmetric matrices are diagonable | |
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| Other results for symmetric matrices | |
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| Spectral decomposition | |
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| Non-negative definite (n.n.d.) matrices | |
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| Simultaneous diagonalization of two symmetric matrices | |
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| The Cayley-Hamilton theorem | |
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| The singular-value decomposition | |
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| Exercises | |
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| Miscellanea | |
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| Orthogonal matrices-a summary | |
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| Idempotent matrices-a summary | |
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| The matrix aI + bJ-a summary | |
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| Non-negative definite matrices-a summary | |
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| Canonical forms and other decompositions-a summary | |
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| Matrix Functions | |
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| Functions of matrices | |
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| Matrices of functions | |
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| Iterative solution of nonlinear equations | |
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| Vectors of differential operators | |
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| Scalars | |
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| Vectors | |
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| Quadratic forms | |
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| Vec and vech operators | |
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| Definitions | |
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| Properties of vec | |
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| Vec-permutation matrices | |
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| Relationships between vec and vech | |
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| Other calculus results | |
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| Differentiating inverses | |
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| Differentiating traces | |
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| Differentiating determinants | |
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| Jacobians | |
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| Aitken's integral | |
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| Hessians | |
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| Matrices with elements that are complex numbers | |
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| Exercises | |
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| Applications in Statistics | |
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| Variance-covariance matrices | |
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| Correlation matrices | |
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| Matrices of sums of squares and cross-products | |
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| Data matrices | |
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| Uncorrected sums of squares and products | |
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| Means, and the centering matrix | |
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| Corrected sums of squares and products | |
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| The multivariate normal distribution | |
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| Quadratic forms and X[superscript 2]-distributions | |
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| Least squares equations | |
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| Contrasts among means | |
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| Exercises | |
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| The Matrix Algebra of Regression Analysis | |
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| General description | |
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| Linear models | |
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| Observations | |
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| Nonlinear models | |
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| Estimation | |
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| Several regressor variables | |
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| Deviations from means | |
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| The statistical model | |
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| Unbiasedness and variances | |
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| Predicted y-values | |
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| Estimating the error variance | |
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| Partitioning the total sum of squares | |
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| Multiple correlation | |
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| The no-intercept model | |
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| Analysis of variance | |
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| Testing linear hypotheses | |
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| Stating a hypothesis | |
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| The F-statistic | |
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| Equivalent statements of a hypothesis | |
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| Special cases | |
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| Confidence intervals | |
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| Fitting subsets of the x-variables | |
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| Reductions in sums of squares: the R([characters not reproducible]) notation | |
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| An Introduction to Linear Statistical Models | |
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| General description | |
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| The normal equations | |
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| A general form | |
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| Many solutions | |
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| Solving the normal equations | |
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| Generalized inverses of X'X | |
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| Solutions | |
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| Expected values and variances | |
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| Predicted y-values | |
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| Estimating the error variance | |
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| Error sum of squares | |
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| Expected value | |
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| Estimation | |
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| Partitioning the total sum of squares | |
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| Coefficient of determination | |
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| Analysis of variance | |
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| The R([characters not reproducible]) notation | |
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| Estimable functions | |
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| Testing linear hypotheses | |
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| Confidence intervals | |
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| Some particular models | |
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| The one-way classification | |
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| Two-way classification, no interactions, balanced data | |
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| Two-way classification, no interactions, unbalanced data | |
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| The R([characters not reproducible]) notation (Continued) | |
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| References | |
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| Index | |