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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 | |