eig

Eigenvalues and eigenvectors

Introduction

Let $A$ be a square matrix. The scalar $\lambda$ and vector $v$ that satisfy the following equation are called eigenvalue and eigenvector, respectively:

An $n\times n$ matrix has $N$ different eigenvalues, where $1\leq N\leq n$. There can be more than one eigenvectors associated with an eigenvalue.

Usages

d = eig(A)

• A is a square matrix.
• It returns the vector d which contains eigenvalues of A.

[V, D] = eig(A)

• A is a square matrix.
• It returns the diagonal matrix D which contains eigenvalues of A.
• Columns of matrix V are the right eigenvectors such that $A\times V=V\times D$.

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

• A is a square matrix.
• It returns the diagonal matrix D that contains eigenvalues of A.
• Columns of matrix V are the right eigenvectors such that $A\times V=V\times D$, and columns of matrix W are the left eigenvectors such that $W'\times A = D\times W'$, where $W'$ is the transpose of $W$.

Examples

• Finding eigenvalues of a random matrix and then group them in conjugate pairs.

Input

a = rand(10);
cplxpair(eig(a))

Output

ans =
-0.4109130 - 0.6592851i
-0.4109130 + 0.6592851i
 0.1072426 - 0.2233978i
 0.1072426 + 0.2233978i
 0.4336722 - 0.6942055i
 0.4336722 + 0.6942055i
-0.3721263 + 0.0000000i
-0.1895444 + 0.0000000i
 0.5893938 + 0.0000000i
 5.6223728 + 0.0000000i

• For the random matrix A, the second eigenvalue is d(2,2) and the corresponding right eigenvector is V(:,2). Here A*V(:,2) and D(2,2)*V(:,2) give approximately the same values.

Input

A = rand(4);
[V,D]=eig(A);
A*V(:,2)
D(2,2)*V(:,2)

Output

ans = 1e-1 ×
-2.3628584 + 1.3160677i
 1.1303095 - 1.2702069i
 1.1515366 + 2.1725013i
-0.0463397 - 1.7423500i

ans = 1e-1 ×
-2.3628584 + 1.3160677i
 1.1303095 - 1.2702069i
 1.1515366 + 2.1725013i
-0.0463397 - 1.7423500i

• For the random matrix A, the second eigenvalue is d(2,2) and the corresponding left eigenvector is W(:,2). Here W(:,2)'*A and D(2,2)*W(:,2)' give approximately the same values.

Input

A = rand(4);
[V,D,W] = eig(A);
W(:,2)'*A
D(2,2)*W(:,2)'

Output

ans = 1e-2 ×
 0.7739085  -0.3822298   0.0919348  -1.4037180

ans = 1e-2 ×
 0.7739085  -0.3822298   0.0919348  -1.4037180