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Basic Matrix Computation

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A Matrix Algebra Approach to Artificial Intelligence

Abstract

In science and engineering, we often encounter the problem of solving a system of linear equations. Matrices provide the most basic and useful mathematical tool for describing and solving such systems. As the introduction to matrix algebra, this chapter presents the basic operations and performance of matrices, followed by a description of vectorization of matrix and matricization of vector.

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References

  1. Banachiewicz, T.: Zur Berechungung der Determinanten, wie auch der Inverse, und zur darauf basierten Auflösung der Systeme linearer Gleichungen. Acta Astron. Sér. C 3, 41–67 (1937)

    MATH  Google Scholar 

  2. Barnett, S.: Matrices: Methods and Applications. Clarendon Press, Oxford (1990)

    MATH  Google Scholar 

  3. Bellman, R.: Introduction to Matrix Analysis, 2nd edn. McGraw-Hill, New York (1970)

    MATH  Google Scholar 

  4. Berberian, S.K.: Linear Algebra. Oxford University Press, New York (1992)

    MATH  Google Scholar 

  5. Boot, J.: Computation of the generalized inverse of singular or rectangular matrices. Am. Math Mon. 70, 302–303 (1963)

    Article  MathSciNet  Google Scholar 

  6. Brewer, J.W.: Kronecker products and matrix calculus in system theory. IEEE Trans. Circuits Syst. 25, 772–781 (1978)

    Article  MathSciNet  Google Scholar 

  7. Brookes, M.: Matrix Reference Manual (2011). https://www.ee.ic.ac.uk/hp/staff/dmb/matrix/intro.html

  8. Duncan, W.J.: Some devices for the solution of large sets of simultaneous linear equations. Lond. Edinb. Dublin Philos. Mag. J. Sci. Ser. 7 35, 660–670 (1944)

    Article  MathSciNet  Google Scholar 

  9. Graybill, F.A.: Matrices with Applications in Statistics. Wadsworth International Group, Balmont (1983)

    Google Scholar 

  10. Graybill, F.A., Meyer, C.D., Painter, R.J.: Note on the computation of the generalized inverse of a matrix. SIAM Rev. 8(4), 522–524 (1966)

    Article  MathSciNet  Google Scholar 

  11. Greville, T.N.E.: Some applications of the pseudoinverse of a matrix. SIAM Rev. 2, 15–22 (1960)

    Article  MathSciNet  Google Scholar 

  12. Guttman, L.: Enlargement methods for computing the inverse matrix. Ann. Math. Stat. 17, 336–343 (1946)

    Article  MathSciNet  Google Scholar 

  13. Henderson, H.V., Searle, S.R.: The vec-permutation matrix, the vec operator and Kronecker products: a review. Linear Multilinear Alg. 9, 271–288 (1981)

    Article  MathSciNet  Google Scholar 

  14. Hendeson, H.V., Searle, S.R.: On deriving the inverse of a sum of matrices. SIAM Rev. 23, 53–60 (1981)

    Article  MathSciNet  Google Scholar 

  15. Horn, R.A., Johnson, C.R.: Matrix Analysis. Cambridge University Press, Cambridge (1985)

    Book  Google Scholar 

  16. Hotelling, H.: Some new methods in matrix calculation. Ann. Math. Stat. 14, 1–34 (1943)

    Article  MathSciNet  Google Scholar 

  17. Hotelling, H.: Further points on matrix calculation and simultaneous equations. Ann. Math. Stat. 14, 440–441 (1943)

    Article  MathSciNet  Google Scholar 

  18. Johnson, L.W., Riess, R.D., Arnold, J.T.: Introduction to Linear Algebra, 5th edn. Prentice-Hall, New York (2000)

    MATH  Google Scholar 

  19. Lancaster, P., Tismenetsky, M.: The Theory of Matrices with Applications, 2nd edn. Academic, New York (1985)

    MATH  Google Scholar 

  20. Lütkepohl, H.: Handbook of Matrices. Wiley, New York (1996)

    MATH  Google Scholar 

  21. Magnus, J.R., Neudecker, H.: The commutation matrix: some properties and applications. Ann. Stat. 7, 381–394 (1979)

    Article  MathSciNet  Google Scholar 

  22. Magnus, J.R., Neudecker, H.: Matrix Differential Calculus with Applications in Statistics and Econometrics, rev. edn. Wiley, Chichester (1999)

    MATH  Google Scholar 

  23. Moore, E.H.: General analysis, part 1. Mem. Am. Philos. Soc. 1, 1 (1935)

    Google Scholar 

  24. Noble, B., Danniel, J.W.: Applied Linear Algebra, 3rd edn. Prentice-Hall, Englewood Cliffs (1988)

    Google Scholar 

  25. Papoulis, A.: Probability, Random Variables and Stochastic Processes. McGraw-Hill, New York (1991)

    Google Scholar 

  26. Penrose, R.A.: A generalized inverse for matrices. Proc. Cambridge Philos. Soc. 51, 406–413 (1955)

    Article  MathSciNet  Google Scholar 

  27. Phillip, A., Regalia, P.A., Mitra, S.: Kronecker products, unitary matrices and signal processing applications. SIAM Rev. 31(4), 586–613 (1989)

    Article  MathSciNet  Google Scholar 

  28. Piegorsch, W.W., Casella, G.: The early use of matrix diagonal increments in statistical problems. SIAM Rev. 31, 428–434 (1989)

    Article  MathSciNet  Google Scholar 

  29. Poularikas, A.D.: The Handbook of Formulas and Tables for Signal Processing. CRC Press, New York (1999)

    MATH  Google Scholar 

  30. Price, C.: The matrix pseudoinverse and minimal variance estimates. SIAM Rev. 6, 115–120 (1964)

    Article  MathSciNet  Google Scholar 

  31. Pringle, R.M., Rayner, A.A.: Generalized Inverse of Matrices with Applications to Statistics. Griffin, London (1971)

    MATH  Google Scholar 

  32. Rado, R.: Note on generalized inverse of matrices. Proc. Cambridge Philos. Soc. 52, 600–601 (1956)

    Article  MathSciNet  Google Scholar 

  33. Rao, C.R., Mitra, S.K.: Generalized Inverse of Matrices and its Applications. Wiley, New York (1971)

    MATH  Google Scholar 

  34. Regalia, P.A., Mitra, S.K.: Kronecker products, unitary matrices and signal processing applications. SIAM Rev. 31(4), 586–613 (1989)

    Article  MathSciNet  Google Scholar 

  35. Schott, J.R.: Matrix Analysis for Statistics. Wiley, New York (1997)

    MATH  Google Scholar 

  36. Searle, S.R.: Matrix Algebra Useful for Statistics. Wiley, New York (1982)

    MATH  Google Scholar 

  37. Sherman, J., Morrison, W.J.: Adjustment of an inverse matrix corresponding to a change in one element of a given matrix. Ann. Math. Stat. 21, 124–127 (1950)

    Article  MathSciNet  Google Scholar 

  38. Woodbury, M.A.: Inverting modified matrices. Memorandum Report 42, Statistical Research Group, Princeton (1950)

    Google Scholar 

  39. Zhang, X.D.: Numerical computations of left and right pseudo inverse matrices (in Chinese). Kexue Tongbao 7(2), 126 (1982)

    Google Scholar 

  40. Zhang, X.D.: Matrix Analysis and Applications. Cambridge University Press, Cambridge (2017)

    Book  Google Scholar 

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Zhang, XD. (2020). Basic Matrix Computation. In: A Matrix Algebra Approach to Artificial Intelligence. Springer, Singapore. https://doi.org/10.1007/978-981-15-2770-8_1

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  • DOI: https://doi.org/10.1007/978-981-15-2770-8_1

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