Problems On Matrices

Problems on Matrices-1:

Example-3 Find the eigen values and eigenvectors of the Matrix A:

With the characteristics equations you get;

With the eigenvalue now the eigenvector can be computed. Therefore you insert the eigenvalue into the matrix and go backward from the matrix to the system of the equations it represents:

So for λ = 2 you will get;

For the second eigenvalue λ = 4 we can find the corresponding eigenvector in the same way:

One solution of this system of the equations is the vector The eigenvector than is every multiple of this vector, so .

Example-4 A matrix A is idempotent if A2 = A. Show that the only possible eigenvalues of an idempotent matrix are λ = 0 and λ = 1. The given an example of amatrix that is idempotent and has both of these two values are eigenvalues.

Solution: Suppose that λ is an eigenvalues of A. Then there is an eigenvector x, such that Ax = λx.

We have

Since x is an eigenvector, it is non zero we get the conclusion that λ² - λ = 0. and the solution to this quadratic polynomial equations λ are λ= 0 and λ = 1. The matrix

is idempotent and since it is a diagonal matrix its eigenvalues are the diagonal enetries λ= 0 and λ = 1.