LINEAR ALGEBRA
AND
APPLIED MATRIX THEORY
by
R.K.S. Rathore
(Ph.D.: I.I.T. Delhi, D.Sc.: Delft University, The Netherlands)
Professor, Department of Mathematics
Indian Institute of Technology, Kanpur
Kanpur – 208016 (UP) India
rksr@iitk.ac.in
Lecture Notes in Mathematics
© R.K.S. Rathore, 609, I.I.T. Kanpur
Contents
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Chapter 1 Fields |
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Links |
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1. |
Groups |
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PROBLEMS |
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2. |
Permutation Groups |
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PROBLEMS |
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3. |
Rings |
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PROBLEMS |
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4. |
Fields |
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PROBLEMS |
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5. |
Vector Spaces and Algebras over a Field |
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PROBLEMS |
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6. |
Polynomials over a Field |
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PROBLEMS |
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7. |
The Division Algorithm |
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PROBLEMS |
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8. |
HCF of Polynomials |
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PROBLEMS |
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9. |
Prime Factorization of Polynomials |
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PROBLEMS |
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10. |
Extension of Fields |
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PROBLEMS |
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Chapter 2 Matrix Theory |
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Links |
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1. |
Matrices |
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PROBLEMS |
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2. |
Matrix Addition |
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PROBLEMS |
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3. |
Matrix Multiplication |
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PROBLEMS |
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4. |
Scalar Multiplication |
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PROBLEMS |
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5. |
Block-Partitioned Matrices and Block Operations |
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PROBLEMS |
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6. |
Vector Spaces of Matrices |
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PROBLEMS |
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7. |
The Standard Inner Product in R n and C n |
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PROBLEMS |
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8. |
Linear Independence of Matrices |
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PROBLEMS |
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9. |
Some Standard Matrices |
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PROBLEMS |
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10. |
Elementary Row and Column Operations |
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PROBLEMS |
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11. |
Row-Reduced Echelon Form |
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PROBLEMS |
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12. |
Determinant of a Square Matrix |
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PROBLEMS |
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13. |
Properties of the Determinant Function |
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PROBLEMS |
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14. |
Cofactor Expansion |
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PROBLEMS |
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15. |
Rank of a Matrix |
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PROBLEMS |
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16. |
The System of Linear Equations: Ax = b |
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PROBLEMS |
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17. |
Eigenvalues and Eigenvectors |
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18. |
Companion Matrices and Characteristic Polynomial |
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19. |
Method of Danilevsky for Characteristic Polynomial |
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PROBLEMS |
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20. |
Matrices with a Full-Set of Eigenvectors |
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PROBLEMS |
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21. |
The Cayley-Hamilton Theorem |
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PROBLEMS |
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22. |
Triangulization and Unitary Diagonalization of a Matrix |
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23. |
Schur’s Lemma and the Spectral Theorem |
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PROBLEMS |
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24. |
The Equation: AX-XB = C |
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PROBLEMS |
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25. |
Circulants and the Discrete Fourier Transform (DFT) |
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PROBLEMS |
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26. |
DFT of a Vector Signal and Solution of a Circulant System |
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PROBLEMS |
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27 |
l -Matrices and Similarity |
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PROBLEMS |
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28. |
Elementary Transformations on l -Matrices |
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PROBLEMS |
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Chapter 3 Canonical Factorizations |
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Links |
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1. |
Row-Reduced Echelon Form |
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2. |
Hermite Canonical Form |
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3. |
Rank Factorization |
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4. |
Rank Factorization (Rectangular Form) |
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5. |
Rank Factorization (Left Invertible and Right Unitary) |
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6. |
Triangular Reduction |
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7. |
Conjugate Diagonal Reduction of Hermitian Matrices |
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8. |
Triangular Reduction by a Unitary Matrix |
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9. |
QR-Decomposition |
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10. |
Gram-Schmidt Triangular Reduction |
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11. |
Simultaneous Reduction of Positive Definite A and Hermitian B |
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12. |
Non-Singular LU-Decomposition |
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PROBLEMS |
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13. |
Cholesky LL * -Decompostion |
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PROBLEMS |
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14. |
Singular Value Decomposition |
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PROBLEMS |
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15. |
Polar Decomposition |
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PROBLEMS |
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16. |
When do the Polar Factors Commute? |
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17. |
Unicity of Polar Decomposition |
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18. |
Equivalence of Polar Decomposition and SVD |
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19. |
To Rotate a Vector in the Direction of another Vector |
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20. |
Tridiagonalizing a Hermitian Matrix by Householder Reflections (in n-2 steps) |
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21. |
QR-Algorithm for Hessenberg Matrices |
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Chapter 4 Vector Spaces |
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Links |
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1. |
Vector Space over a Field |
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PROBLEMS |
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2. |
Linear Independence of Vectors |
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PROBLEMS |
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3. |
Bases in a Vector Space |
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PROBLEMS |
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4. |
Dimension of a Vector Space |
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PROBLEMS |
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5. |
Direct Sum Decomposition of a Vector Space |
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PROBLEMS |
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6. |
Linear Transformations (Operators) |
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PROBLEMS |
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7. |
Change of Bases |
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PROBLEMS |
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8. |
Jordan Canonical Form |
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PROBLEMS |
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9. |
Rank of a Linear Transformation |
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PROBLEMS |
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10. |
Linear Functionals |
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PROBLEMS |
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11. |
The Transpose of a Linear Transformation |
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PROBLEMS |
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12. |
Invariant Subspaces and Direct Sum of Operators |
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PROBLEMS |
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13. |
Norms on Finite Dimensional Real or Complex Vector Spaces |
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PROBLEMS |
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14. |
The Hölder and Minkowski’s Inequalities |
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15. |
Equivalence of Norms on Finite Dimensional Spaces |
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16. |
Matrix Norms |
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PROBLEMS |
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17. |
Inner Product Spaces |
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PROBLEMS |
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18. |
Norm Induced by an Inner Product |
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PROBLEMS |
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19. |
Orthogonality in Inner Product Spaces |
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PROBLEMS |
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20. |
The Angle Between Two Vectors |
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PROBLEMS |
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21. |
Gram-Schmidt Orthonormalization Procedure |
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PROBLEMS |
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22. |
Projections and Orthogonal Projections |
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PROBLEMS |
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23. |
The Kaczmarz Method |
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24. |
The Method of Residual Projections |
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Chapter 5 Second Order Forms |
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Links |
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1. |
Bilinear and Multilinear Expressions on Product Spaces |
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PROBLEMS |
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2. |
Bilinear Forms on a Vector Space |
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PROBLEMS |
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3. |
Symmetric and Skew-Symmetric Bilinear Forms |
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PROBLEMS |
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4. |
Quadratic Forms |
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PROBLEMS |
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5. |
Sesqi-Linear and Hermitian Forms |
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PROBLEMS |
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6. |
Characterization of a Positive Definite Matrix |
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PROBLEMS |
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Chapter 6 Simultaneous Triangulization and Diagonalization |
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Links |
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1. |
Characteristic and Minimal Polynomials |
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PROBLEMS |
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2. |
T-Invariant Subspaces and T-Conductors |
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PROBLEMS |
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3. |
Triangulization and Diagonalization |
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PROBLEMS |
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4. |
Simultaneous Triangulization and Diagonalization |
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PROBLEMS |
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5. |
Simultaneous Diagonalization |
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PROBLEMS |
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Chapter 7 The Primary and The Cyclic Decomposition Theorems |
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Links |
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1. |
The Primary Decomposition Theorem |
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PROBLEMS |
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2. |
T-Cyclic Subspaces and Vectors |
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PROBLEMS |
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3. |
The Cyclic Decomposition Theorem |
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PROBLEMS |
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4. |
The Rational Form |
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PROBLEMS |
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5. |
The Invariant Factors |
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PROBLEMS |
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6. |
The Jordan Canonical Form |
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PROBLEMS |
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7. |
A Rank Based Determination of the Jordan Canonical Form |
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PROBLEMS |
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Chapter 8 Generalized Inverses |
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Links |
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1. |
Definition of a g-Inverse |
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PROBLEMS |
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2. |
Reflexive g-Inverse A-r |
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PROBLEMS |
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3. |
Least-Squares g-Inverse A-r |
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PROBLEMS |
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4. |
Minimum Norm g-Inverse A-m |
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PROBLEMS |
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5. |
Moore-Penrose Inverse A+ |
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PROBLEMS |
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Chapter 9 Courant-Fischer Min-Max Theorems |
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Links |
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1. |
A Variational Characterization of Eigen Values |
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2. |
Given’s Method for a Hermitian Matrix |
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3. |
Eigenvalues of a Tridiagonal Hermitian Matrix |
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Chapter 10 Inequalities in Matrix Theory |
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Links |
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1. |
The Volume of an m-Dimensional Parallelepiped |
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2. |
Change of Variables in Multiple Integrals |
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3. |
Surface and Volume of a Hyper-Sphere |
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4. |
Volume of an n-Dimensional Ellipsoid |
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5. |
Principal Semi-Axes of a Section of an Ellipsoid |
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6. |
Application of Poincaré Separation Theorem to Ellipsoids |
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Chapter 11 Perron-Frobenius Theory of Non-Negative Matrices |
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Links |
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1. |
Non-Negative Matrices and Vectors |
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PROBLEMS |
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2. |
Perron’s Theorem on Positive Matrices |
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PROBLEMS |
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3. |
Irreducible Matrices and Directed Graphs |
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PROBLEMS |
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4. |
The Perron-Frobenius Theorem |
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PROBLEMS |
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5. |
The Structure of Cyclic Matrices |
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PROBLEMS |
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Chapter 12 Stability of a Matrix |
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Links |
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PROBLEMS |
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1. |
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PROBLEMS |
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2. |
Equivalent Formulations of Sylvester’s Inertia Theorem |
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PROBLEMS |
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3. |
Some Equivalent Criteria of Stability |
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