Eigenvalues and Eigenvectors in MATLAB

Eigenvalues and Eigenvectors are properties of a square matrix.

Let is an N*N matrix, X be a vector of size N*1 and  be a scalar.

Then the values X,  satisfying the equation    are eigenvectors and eigenvalues of matrix A respectively.

• A matrix of size N*N possess N eigenvalues
• Every eigenvalue corresponds to an eigenvector.

Matlab allows the users to find eigenvalues and eigenvectors of matrix using eig() method. Different syntaxes of eig() method are:

• e = eig(A)
• [V,D] = eig(A)
• [V,D,W] = eig(A)
• e = eig(A,B)

Let us discuss the above syntaxes in detail:

e = eig(A)

• It returns the vector of eigenvalues of square matrix A.

Output :

[V,D] = eig(A)

• It returns the diagonal matrix D having diagonals as eigenvalues.
• It also returns the matrix of right vectors as V.
• Normal eigenvectors are termed as right eigenvectors.
• V is a collection of N eigenvectors of each N*1 size(A is N*N size) that satisfies A*V = V*D

Output :

[V,D,W] = eig(A)

• Along with the diagonal matrix of eigenvalues D and right eigenvectors V, it also returns the left eigenvectors of matrix A.
• A left eigenvector u is a 1*N matrix that satisfies the equation u*A = k*u, where k is a left eigenvalue of matrix A.
• W is the collection of N left eigenvectors of A that satisfies W'*A = D*W'.

Output :

e = eig(A,B)

• It returns the generalized eigenvalues of two square matrices A and B of the same size.
• A generalized eigenvalue λ and a corresponding eigenvector v satisfy Av=λBv.

Output :