Eigenvalues and Eigenvectors - Linear Algebra

An eigenvalue of an involutory matrix must be either 1 or . This can be seen as follows:

Let be an eigenvalue of involutory matrix . Then for some eigenvector ,

Premultiply both sides by :

By definition, an involutory matrix has as its square, so

By transitivity,

Thus, , or

It follows that . The statement is false.

The first step into solving for eigenvalues, is adding in a along the main diagonal.

Now the next step to take the determinant.

Now lets FOIL, and solve for .

Now lets use the quadratic equation to solve for .

So our eigen values are

The eigenvalues, , for the matrix are values for which the determinant of is equal to zero. First, find the determinant:

Now set the determinant equal to zero and solve this quadratic:

this can be factored:

The eigenvalues are 5 and 1.

To determine if something is an eignevector, multiply times A:

Since this is equivalent to , is an eigenvector (and 5 is an eigenvalue).

This cannot be re-written as times a scalar, so this is not an eigenvector.

The eigenvalues are scalar quantities, , where the determinant of is equal to zero.

First, find an expression for the determinant:

Now set this equal to zero, and solve:

this can be factored (or solved in another way)

The eigenvalues are -5 and 3.

To determine if something is an eigenvector, multiply by the matrix A:

This is equivalent to so this is an eigenvector.

This is equivalent to so this is also an eigenvector.

The eigenvalues are scalar quantities where the determinant of is equal to zero. First, write an expression for the determinant:

this can be solved by factoring:

The solutions are -2 and -7

To determine if a vector is an eigenvector, multiply with A:

. This cannot be expressed as an integer times , so is not an eigenvector

This can be expressed as , so is an eigenvector.