Abstract

An iterative method for finding the eigenvectors and eigenvalues of a matrix via incoherent optical matrix-vector multiplication and simple electronic feedback is described.

© 1981 Optical Society of America

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References

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  1. J. W. Goodman, A. R. Dias, L. M. Woody, Opt. Lett. 2, 1 (1978).
    [CrossRef] [PubMed]
  2. J. W. Goodman, A. R. Dias, L. M. Woody, L. Erickson, Proc. Soc. Photo-Opt. Instrum. Eng.485 (1979).
  3. D. Psaltis, D. Casasent, M. Carlotto, Opt. Lett. 4, 348 (1979).
    [CrossRef] [PubMed]
  4. J. H. Wilkinson, The Algebraic Eigenvalue Problem (Clarendon, Oxford, 1965).
  5. D. M. Young, R. T. Gregory, A Survey Of Numerical Mathematics II (Addison-Wesley, Reading, Mass., 1973).
  6. G. Strang, Linear Algebra and Its Applications (Academic, New York, 1976), p. 175.

1979 (2)

J. W. Goodman, A. R. Dias, L. M. Woody, L. Erickson, Proc. Soc. Photo-Opt. Instrum. Eng.485 (1979).

D. Psaltis, D. Casasent, M. Carlotto, Opt. Lett. 4, 348 (1979).
[CrossRef] [PubMed]

1978 (1)

Carlotto, M.

Casasent, D.

Dias, A. R.

J. W. Goodman, A. R. Dias, L. M. Woody, L. Erickson, Proc. Soc. Photo-Opt. Instrum. Eng.485 (1979).

J. W. Goodman, A. R. Dias, L. M. Woody, Opt. Lett. 2, 1 (1978).
[CrossRef] [PubMed]

Erickson, L.

J. W. Goodman, A. R. Dias, L. M. Woody, L. Erickson, Proc. Soc. Photo-Opt. Instrum. Eng.485 (1979).

Goodman, J. W.

J. W. Goodman, A. R. Dias, L. M. Woody, L. Erickson, Proc. Soc. Photo-Opt. Instrum. Eng.485 (1979).

J. W. Goodman, A. R. Dias, L. M. Woody, Opt. Lett. 2, 1 (1978).
[CrossRef] [PubMed]

Gregory, R. T.

D. M. Young, R. T. Gregory, A Survey Of Numerical Mathematics II (Addison-Wesley, Reading, Mass., 1973).

Psaltis, D.

Strang, G.

G. Strang, Linear Algebra and Its Applications (Academic, New York, 1976), p. 175.

Wilkinson, J. H.

J. H. Wilkinson, The Algebraic Eigenvalue Problem (Clarendon, Oxford, 1965).

Woody, L. M.

J. W. Goodman, A. R. Dias, L. M. Woody, L. Erickson, Proc. Soc. Photo-Opt. Instrum. Eng.485 (1979).

J. W. Goodman, A. R. Dias, L. M. Woody, Opt. Lett. 2, 1 (1978).
[CrossRef] [PubMed]

Young, D. M.

D. M. Young, R. T. Gregory, A Survey Of Numerical Mathematics II (Addison-Wesley, Reading, Mass., 1973).

Opt. Lett. (2)

Proc. Soc. Photo-Opt. Instrum. Eng. (1)

J. W. Goodman, A. R. Dias, L. M. Woody, L. Erickson, Proc. Soc. Photo-Opt. Instrum. Eng.485 (1979).

Other (3)

J. H. Wilkinson, The Algebraic Eigenvalue Problem (Clarendon, Oxford, 1965).

D. M. Young, R. T. Gregory, A Survey Of Numerical Mathematics II (Addison-Wesley, Reading, Mass., 1973).

G. Strang, Linear Algebra and Its Applications (Academic, New York, 1976), p. 175.

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Figures (1)

Fig. 1
Fig. 1

Heart of the eigenvector analysis device is the optical matrix multiplier.1,2 Analog circuitry provides the required feedback.

Equations (22)

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| λ 1 | > | λ 2 | > > | λ N 1 | > | λ N | .
V ( 0 ) = c 1 e 1 + + c N e N .
V ( 1 ) = c 1 λ 1 e 1 + + c N λ N e N .
V ( n ) = M V ( n 1 ) = c 1 λ 1 n e 1 + + c N λ N n e N .
V ( n ) c 1 λ 1 n e 1 .
V ( n + 1 ) c 1 λ 1 n + 1 e 1 ,
V ( n + 1 ) λ 1 V ( n )
W ( n + 1 ) λ 1 U ( n ) .
M = ( m 11 m 1 N m N 1 m N N )
M = M + M ,
m m n + = 0 if m m n 0 m m n = 0 if m m n 0 .
V ( k ) = V ( k ) + V ( k ) .
V ( k + 1 ) = M V ( k )
V ( k + 1 ) + V ( k + 1 ) = ( M + M ) [ V ( k + 1 ) + V ( k + 1 ) ] = [ M + V ( k + 1 ) + + M V ( k + 1 ) ] [ M + V ( k + 1 ) + M V ( k + 1 ) + ] .
V ( k + 1 ) + = M + V ( k + 1 ) + + M V ( k + 1 ) ,
V ( k + 1 ) = M + V ( k + 1 ) + M V ( k + 1 ) + .
y ( k ) = [ V ( k ) + V ( k ) ] ,
B = [ M + M M M + ]
y ( k + 1 ) = B y ( k ) = [ V ( k + 1 ) + V ( k + 1 ) ] .
M k = n = 1 k 1 ( M λ n I ) .
M k e = [ n = 1 k 1 ( λ λ n ) ] e .
n = 1 k 1 ( λ λ n ) = Λ k .

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