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The usual number of scalar operations (i.e., the total number of additions and multiplications) required to
perform Matrix Multiplication is
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(1) |
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(2) |
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|
(3) | |||
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|
(4) |
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(5) |
Two matrices can therefore be multiplied
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(6) |
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(7) |
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(8) |
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(9) |
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(10) |
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(11) |
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(12) |
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(13) |
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(14) |
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(15) |
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(16) |
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(17) |
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(18) |
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(19) |
Matrix inversion of a matrix
to yield
can also be done in fewer operations
than expected using the formulas
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(20) |
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(21) |
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(22) |
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(23) |
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(24) |
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(25) |
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(26) |
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(27) |
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(28) |
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(29) |
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(30) |
See also Complex Multiplication, Karatsuba Multiplication
References
Coppersmith, D. and Winograd, S. ``Matrix Multiplication via Arithmetic Programming.'' J. Symb. Comput. 9, 251-280, 1990.
Pan, V. How to Multiply Matrices Faster. New York: Springer-Verlag, 1982.
Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetterling, W. T. ``Is Matrix Inversion an
Strassen, V. ``Gaussian Elimination is Not Optimal.'' Numerische Mathematik 13, 354-356, 1969.
Process?''
§2.11 in Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed.
Cambridge, England: Cambridge University Press, pp. 95-98, 1989.
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© 1996-9 Eric W. Weisstein