=eig(A) returns diagonal matrix D ofeigenvalues and matrix V whose columns are thecorresponding right eigenvectors, so that A*V = V*D.

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= eig(A) also returns full matrix W whosecolumns are the corresponding left eigenvectors, so that W"*A= D*W".

The eigenvalue problem is to determine the solution to the equation Av = λv,where A is an n-by-n matrix, v isa column vector of length n, and λ isa scalar. The values of λ that satisfy theequation are the eigenvalues. The corresponding values of v thatsatisfy the equation are the right eigenvectors. The left eigenvectors, w,satisfy the equation wA = λw’.

e = eig(A,B) returnsa column vector containing the generalized eigenvalues of square matrices A and B.

=eig(A,B) returnsdiagonal matrix D of generalized eigenvalues andfull matrix V whose columns are the correspondingright eigenvectors, so that A*V = B*V*D.

= eig(A,B) alsoreturns full matrix W whose columns are the correspondingleft eigenvectors, so that W"*A = D*W"*B.

The generalized eigenvalue problem is to determine the solutionto the equation Av = λBv,where A and B are n-by-n matrices, v isa column vector of length n, and λ isa scalar. The values of λ that satisfy theequation are the generalized eigenvalues. The corresponding valuesof v are the generalized right eigenvectors. Theleft eigenvectors, w, satisfy the equation wA = λwB.

<___> = eig(A,balanceOption), where balanceOption is "nobalance", disables the preliminary balancing step in the algorithm. The default for balanceOption is "balance", which enables balancing. The eig function can return any of the output arguments in previous syntaxes.

<___> = eig(A,B,algorithm),where algorithm is "chol", usesthe Cholesky factorization of B to compute thegeneralized eigenvalues. The default for algorithm dependson the properties of A and B,but is generally "qz", which uses the QZ algorithm.

If A is Hermitian and B isHermitian positive definite, then the default for algorithm is "chol".

<___> = eig(___,outputForm) returns the eigenvalues in the form specified by outputForm using any of the input or output arguments in previous syntaxes. Specify outputForm as "vector" to return the eigenvalues in a column vector or as "matrix" to return the eigenvalues in a diagonal matrix.

A = 4×4 1.0000 0.5000 0.3333 0.2500 0.5000 1.0000 0.6667 0.5000 0.3333 0.6667 1.0000 0.7500 0.2500 0.5000 0.7500 1.0000
V = 3×3 complex -0.5774 + 0.0000i 0.5774 + 0.0000i 0.5774 + 0.0000i -0.5774 + 0.0000i -0.2887 - 0.5000i -0.2887 + 0.5000i -0.5774 + 0.0000i -0.2887 + 0.5000i -0.2887 - 0.5000i
D = 3×3 complex 6.0000 + 0.0000i 0.0000 + 0.0000i 0.0000 + 0.0000i 0.0000 + 0.0000i -1.5000 + 0.8660i 0.0000 + 0.0000i 0.0000 + 0.0000i 0.0000 + 0.0000i -1.5000 - 0.8660i
ans = 3×3 complex10-14 × -0.2665 + 0.0000i -0.0888 - 0.0111i -0.0888 + 0.0111i 0.0888 + 0.0000i 0.0000 + 0.0833i 0.0000 - 0.0833i -0.0444 + 0.0000i -0.1157 + 0.0666i -0.1157 - 0.0666i
Ideally, the eigenvalue decomposition satisfies the relationship. Since eig performs the decomposition using floating-point computations, then A*V can, at best, approach V*D. In other words, A*V - V*D is close to, but not exactly, 0.

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By default eig does not always return the eigenvalues and eigenvectors in sorted order. Use the sort function to put the eigenvalues in ascending order and reorder the corresponding eigenvectors.