To find the eigenvectors of a matrix, we need to first find the eigenvalues. Here are the steps to find eigenvectors:

  1. Start by finding the eigenvalues of the matrix. An eigenvalue is a scalar λ that satisfies the equation Av = λv, where A is the matrix and v is the eigenvector.

  2. To find the eigenvalues, we solve the characteristic equation det(A - λI) = 0, where A is the matrix, λ is the eigenvalue, and I is the identity matrix of the same size as A.

  3. Solving the characteristic equation will give us a set of eigenvalues for the matrix.

  4. For each eigenvalue, we find the corresponding eigenvector by solving the equation (A - λI)v = 0, where A is the matrix, λ is the eigenvalue, and v is the eigenvector.

  5. To solve this equation, we create a matrix (A - λI) and set it up as a system of linear equations. We then perform row operations to convert the matrix into row echelon form or reduced row echelon form. After that, we can determine the values of the***** variables to obtain the eigenvectors.

  6. The eigenvectors corresponding to each eigenvalue will form a set of linearly independent vectors that span the eigenspace.

  It's important to note that not all matrices have eigenvectors. In some cases, matrices may have complex eigenvalues and eigenvectors.

  I hope this explanation helps you understand how to find eigenvectors of a matrix. If you have any further questions, feel***** to ask!