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Eigenvalues and eigenvectors

Let's imagine an arbitrary real n×n matrix, A. It is very possible that when we apply this matrix to some vector, they are scaled by a constant value. If this is the case, we say that the nonzero -dimensional vector is an eigenvector of A, and it corresponds to an eigenvalue λ. We write this as follows:

Note: The zero vector (0) cannot be an eigenvector of A, since A0 = 0 = λ0 for all λ.

Let's consider again a matrix A that has an eigenvector x and a corresponding eigenvalue λ. Then, the following rules will apply:

If we have a matrix A and it has been shifted from its current position to , then it has the eigenvector x and the corresponding eigenvalue , for all , so that .
If the matrix A is invertible, then x is also an eigenvector of the inverse of the matrix, , with the corresponding eigenvalue .
for any .

We know from earlier in the chapter that whenever we multiply a matrix and a vector, the direction of the vector is changed, but this is not the case with eigenvectors. They are in the same direction as A, and thus x remains unchanged. The eigenvalue, being a scalar value, tells us whether the eigenvector is being scaled, and if so, how much, as well as if the direction of the vector has changed.

Another very fascinating property the determinant has is that it is equivalent to the product of the eigenvalues of the matrix, and it is written as follows:

But this isn't the only relation that the determinant has with eigenvalues. We can rewrite in the form. And since this is equal to zero, this means it is a non-invertible matrix, and therefore its determinant too must be equal to zero. Using this, we can use the determinant to find the eigenvalues. Let's see how.

Suppose we have . Then, its determinant is shown as follows:

We can rewrite this as the following quadratic equation:

We know that the quadratic equation will give us both the eigenvalues . So, we plug our values into the quadratic formula and get our roots.

Another interesting property is that when we have triangular matrices such as the ones we found earlier in this chapter, their eigenvalues are the pivot values. So, if we want to find the determinant of a triangular matrix, then all we have to do is find the product of all the entries along the diagonal.