Eigenvalue Decomposition


Eigenvalues and eigenvectors of a real matrix.

If A is symmetric, then A = V*D*V' where the eigenvalue matrix D is diagonal and the eigenvector matrix V is orthogonal. I.e. A = V*D*V' and V*V' equals the identity matrix.

If A is not symmetric, then the eigenvalue matrix D is block diagonal with the real eigenvalues in 1-by-1 blocks and any complex eigenvalues, lambda + i*mu, in 2-by-2 blocks, [lambda, mu; -mu, lambda]. The columns of V represent the eigenvectors in the sense that A*V = V*D. The matrix V may be badly conditioned, or even singular, so the validity of the equation A = V*D*MatInverse(V) depends upon MatCond(V)

eig = MatEig(a())

- This function returns an object which contains Eigenvalues and eigenvectors of a real matrix. The following funcions return the results of the decomposition.



c() = MatEigReal(eig)

- Return the real parts of the eigenvalues.

c() = MatEigImaginary(eig)

- Return the imaginary parts of the eigenvalues.

c() = MatEigVector(eig)

- Return the eigenvector matrix.

c() = MatEigBlockDiagonal(eig)

- Return the block diagonal eigenvalue matrix.



Linear Algebra







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