How Do You Find the Eigenvectors of a Matrix?

You find the eigenvectors of a matrix by following these steps:

1. Find the eigenvalues: Calculate the eigenvalues of the matrix by solving the characteristic equation, which is det(A - λI) = 0, where A is the matrix, λ is the eigenvalue, and I is the identity matrix.
2. Solve for the eigenvectors: For each eigenvalue, substitute it back into the equation (A - λI)v = 0, where v is the eigenvector. Solve this system of linear equations to find the eigenvector corresponding to that eigenvalue.

Example

Let's say we have the following matrix:

A = [[2, 1],
     [1, 2]]

1. Find the eigenvalues:

   • The characteristic equation is: det(A - λI) = det([[2, 1], [1, 2]] - λ[[1, 0], [0, 1]]) = (2 - λ)(2 - λ) - 1 = λ² - 4λ + 3 = 0.
   • Solving for λ, we get λ = 1 and λ = 3.

2. Solve for the eigenvectors:

   • For λ = 1:
     • (A - λI)v = [[1, 1], [1, 1]]v = 0.
     • Solving this system, we get v = [1, -1].
   • For λ = 3:
     • (A - λI)v = [[-1, 1], [1, -1]]v = 0.
     • Solving this system, we get v = [1, 1].

Therefore, the eigenvectors of the matrix A are [1, -1] and [1, 1] corresponding to the eigenvalues 1 and 3, respectively.

Practical Insights

• Eigenvectors represent the directions in which a matrix scales or transforms vectors.
• The eigenvalues indicate the scaling factor along these directions.
• Eigenvectors and eigenvalues are crucial in various applications, including linear algebra, differential equations, and quantum mechanics.