Author | Hart, George W. author |
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Title | Multidimensional Analysis [electronic resource] : Algebras and Systems for Science and Engineering / by George W. Hart |
Imprint | New York, NY : Springer New York, 1995 |
Connect to | http://dx.doi.org/10.1007/978-1-4612-4208-6 |
Descript | XII, 236 p. online resource |
0. Introductory -- 0.1 Physical Dimensions -- 0.2 Mathematical Dimensions -- 0.3 Overview -- Exercises -- 1. The Mathematical Foundations of Science and Engineering -- 1.1 The Inadequacy of Real Numbers -- 1.2 The Mathematics of Dimensioned Quantities -- 1.3 Conclusions -- Exercises -- 2. Dimensioned Linear Algebra -- 2.1 Vector Spaces and Linear Transformations -- 2.2 Terminology and Dimensional Inversion -- 2.3 Dimensioned Scalars -- 2.4 Dimensioned Vectors -- 2.5 Dimensioned Matrices -- Exercises -- 3. The Theory of Dimensioned Matrices -- 3.1 The Dimensional Freedom of Multipliable Matrices -- 3.2 Endomorphic Matrices and the Matrix Exponential -- 3.3 Square Matrices, Inverses, and the Determinant -- 3.4 Squarable Matrices and Eigenstructure -- 3.5 Dimensionally Symmetric Multipliable Matrices -- 3.6 Dimensionally Hankel and Toeplitz Matrices -- 3.7 Uniform, Half Uniform, and Dimensionless Matrices -- 3.8 Conclusions -- 3.A Appendix: The n + m ? 1 Theorem -- Exercises -- 4. Norms, Adjoints, and Singular Value Decomposition -- 4.1 Norms for Dimensioned Spaces -- 4.2 Dimensioned Singular Value Decomposition (DSVD) -- 4.3 Adjoints -- 4.4 Norms for Nonuniform Matrices -- 4.5 A Control Application -- 4.6 Factorization of Symmetric Matrices -- Exercises -- 5. Aspects of the Theory of Systems -- 5.1 Differential and Difference Equations -- 5.2 State-Space Forms -- 5.3 Canonical Forms -- 5.4 Transfer Functions and Impulse Responses -- 5.5 Duals and Adjoints -- 5.6 Stability -- 5.7 Controllability, Observability, and Grammians -- 5.8 Expectations and Probability Densities -- Exercises -- 6. Multidimensional Computational Methods -- 6.1 Computers and Engineering -- 6.2 Representing and Manipulating Dimensioned Scalars -- 6.3 Dimensioned Vectors -- 6.4 Representing Dimensioned Matrices -- 6.5 Operations on Dimensioned Matrices -- 6.6 Conclusions -- Exercises -- 7. Forms of Multidimensional Relationships -- 7.1 Goals -- 7.2 Operations -- 7.3 Procedure -- Exercises -- 8. Concluding Remarks -- 9. Solutions to Odd-Numbered Exercises -- References