9. Eigenvalues and eigenvectors.

A linear operator L on the vector space V can admit a set of vectors which, when transformed by L, result multiplied by a scalar.

Vectors of this set are said eigenvectors li and the corresponding scalars are said eigenvalues λi of the linear operator L.

To compute eigenvectors and eigenvalues of L one need to solve the following equation said eigenvalue equation


From this equation we have


The last equation allows the calculation of the eigenvalues λi. It is easy if the dimension n of the vector space is 2, but difficult if n is greater, because it is equivalent to a polynomial equation of degree n and, in general, it can have complex roots.

However if the matrix of L is diagonal, that is if all its entries are null except those with equal indices, the equation is equivalent to Eqn903.gif with a trivial set of solutions Eqn904.gif.

In this case, the eigenvalue equation is equivalent to the set of equations


which admit solution only if the components of the eigenvector |li> are null except that with index i.


Given the diagonal matrix Eqn906.gif, one has Eqn907.gif that is Eqn908.gif therefore Eqn909.gif.

The eigenvector |d1>, corresponding to the eigenvalue δ1=a, is obtained from the equation Eqn910.gif, that is Eqn911.gif, true for all x if y=0. So the calculation doesn't find a determined vector, but the set of all the vectors laying on the x-axis. In order to determine the result we select from this set the vector with unitary norm and with the same direction as the x-axis. This is the normalized eigenvector. So in the example we have Eqn912.gif.

The analogous calculation with respect to the eigenvalue |d2>, corresponding to the eigenvalue δ2=2, we obtain Eqn913.gif.


The results shown in the above example can be generalized to n-dimensional diagonal matrices. Therefore, in general:

From the eigenvalue equation we can deduce that, given the matrix L and another matrix M with its inverse M-1, the matrix M-1LM has the same eigenvalues as L. In fact


In particular this is true if M is a rotation.

If the matrix which undergoes the rotation M is a diagonal real matrix like the D of the example, the transformed matrix S=M-1DM is a symmetric real matrix, that is such that, if i≠j, Si,j=Sj,i. In fact we have


Vice versa, using the opposite procedure, every real symmetric matrix can be transformed in a diagonal real matrix.

In fact, given the matrix Eqn928.gif, in order to find the rotation which diagonalizes S, we let


The term by term subtraction of the second equation from the third one gives


From this equation and from the first one we have


Therefore the rotation with angle -θ transforms S into D.

This procedure can be applied to any symmetric real matrix, so we can say that all the eigenvalues of a real symmetric matrix are real values.


Given the real symmetric matrix Eqn916.gif the eigenvalue equation is Eqn917.gif from which we have


The eigenvalues are σ1=4 e σ2=8. In order to calculate the corresponding normalized eigenvectors one need to solve the equations Eqn919.gif, where the norm of the vectors |si> is 1, so they can be expressed as Eqn920.gif with Eqn922.gif.

Using the first eigenvalue we have


therefore Eqn924.gif.

Using the second eigenvalue we have


therefore Eqn926.gif.

The eigenvectors are orthonormal.


There exists a rotation that transforms S into a diagonal matrix with the same eigenvalues Eqn927.gif. The same rotation transforms the normalized eigenvectors of S into the unit vectors of the cartesian axes. In the proposed example the rotation angle is π/6.

Eigenvalues and eigenvectors with WolframAlpha