So, you've cracked the code and found your eigenvalues. Awesome! But the journey doesn't end there. Now, let's learn how to compute eigenvectors from eigenvalues. Think of eigenvalues as the 'energy levels' of a system and eigenvectors as the 'shapes' associated with those energy levels. They describe how a linear transformation stretches or compresses space.
Here's the gist: For each eigenvalue (λ), you'll solve the equation (A - λI)v = 0, where A is your original matrix, I is the identity matrix, and v is the eigenvector you're trying to find. This results in a system of linear equations.
Solve for the variables in 'v'. You'll often find that the solution isn't unique – that's perfectly fine! Eigenvectors are defined up to a scalar multiple. Express your eigenvector in terms of free variables, and you've got it!
In essence, finding eigenvectors boils down to solving a system of linear equations for *each* eigenvalue. Once you master this, you'll truly understand the power of eigenvalues and eigenvectors in analyzing linear transformations.
