Abstract: The authors present their analysis of the differential equation dX(t)/dt=AX(t)-X^T(t)BX(t)X(t), where A is an unsymmetrical real matrix, B is a positive definite symmetric real matrix, X∈Rⁿ; showing that the equation characterizes a class of continuous type full-feedback artificial neural network; We give the analytic expression of the solution; discuss its asymptotic behavior; and finally present the result showing that, in almost all cases, one and only one of following cases is true. 1. For any initial value X₀∈Rⁿ, the solution approximates asymptotically to zero vector. In this case, the real part of each eigenvalue of A is non-positive. 2. For any initial value X₀ outside a proper subspace of Rⁿ, the solution approximates asymptotically to a nontrivial constant vector &Ytilde;(X₀). In this case, the eigenvalue of A with maximal real part is the positive number λ=‖(X₀)‖²_B and (X₀) is the corresponding eigenvector. 3. For any initial value X₀ outside a proper subspace of Rⁿ, the solution approximates asymptotically to a non-constant periodic function &Ytilde;(X₀,t). Then the eigenvalues of A with maximal real part is a pair of conjugate complex numbers which can be computed.

Key words: PCA, Unsymmetrical real matrix, Eigenvalue, Eigenvector, Neural network

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