# Eigenvalues and Eigenvectors in MATLAB

Last Updated : 20 Nov, 2021

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.

## Matlab

 % Square matrix of size 3*3 A = [0 1 2;     1 0 -1;     2 -1 0]; disp("Matrix"); disp(A);   % Eigenvalues of matrix A e = eig(A); disp("Eigenvalues"); disp(e);

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

## Matlab

 % Square matrix of size 3*3 A = [8 -6 2;     -6 7 -4;     2 -4 3]; disp("Matrix"); disp(A);   % Eigenvalues and right eigenvectors of matrix A [V,D] = eig(A); disp("Diagonal matrix of Eigenvalues"); disp(D); disp("Right eigenvectors") disp(V);

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’.

## Matlab

 % Square matrix of size 3*3 A = [10 -6 2;     -6 7 -4;      2 -4 3]; disp("Matrix :"); disp(A);   % Eigenvalues and right and left eigenvectors  % of matrix A [V,D,W] = eig(A); disp("Diagonal matrix of Eigenvalues :"); disp(D); disp("Right eigenvectors :") disp(V); disp("Left eigenvectors :") disp(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 Î» and a corresponding eigenvector v satisfy Av=Î»Bv.

## Matlab

 % Square matrix A and B of size 3*3 A = [10 -6 2;     -6 7 -4;      2 -4 3]; B = [8 6 1;      6 17 2;     -1 4 3];       disp("Matrix A:"); disp(A); disp("Matrix B:"); disp(B);   % Generalized eigen values  % of matrices A and B e = eig(A,B); disp("Generalized eigenvalues :") disp(e);

Output :

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