Calculation of Eigenvalues in Linear Algebra

The way to go about calculating these quantities, is you first calculate eigenvalues, and then use those values to calculate the eigenvectors.

Calculating eigenvalues

If you recall, the eigenvector-eigenvalue equation was:

$Ax=\lambda x$

This equation can be rewritten as:

$(A-\lambda I)x=0$

The specific proof of this outside the scope of the current course. If you're interested, we have included a full mathematical derivation of this result in the Additional Reading section.

From this, it follows that the equation for calculating eigenvalues is:

$det(A-\lambda I)=0$

Calculating Eigenvectors

Once you have calculated eigenvalues, we then substitute that in the original equation to find the eigenvectors.

$(A-\lambda I)x=0$

This calculation might get tricky, because owing to the nature of eigenvectors, the two equations you will get will be redundant. For this case, let's eigenvector x is (x1, x2). On solving these, we first find a relationship between x1 and x2 (for example, x1 = -3x2), and then substitute the simplest value.

Ideally, we want to express eigenvectors with unit length. To do this, divide each component of the vector by the magnitude of the vector.