Study Notes on Eigen Values and Eigen Vectors

Eigen values and Eigenvectors of Matrix

Let A be a n×n matrix.

λ is an Eigen value for A if there exists a vector u≠0 such that

Au = λu

If such a vector u exists, it is said to be an eigenvector associated with the Eigen value λ.

Suppose that λ is an Eigen value for A.

Then there exists a non-zero vector  such that Au = λu or, equivalently, such that (A - λI_n)u = 0, where I_n is the n×n identity matrix.

Since (A - λI_n)u = 0 has a non-trivial solution u, the matrix (A - λI_n) is not invertible, i.e. |A - λI_n| = 0.

Note that |A - λI_n| is a polynomial in λ, so you get the eigenvalues of A by finding the roots of |A - λI_n|

Say λ₀ is such an eigenvalue. In order to find the eigenvectors associated with λ₀, you have to solve the system Au = λ₀u for x₁, …, x_n.

Example

To find the Eigen values and Eigen vectors of two by two matrices

Let 

To find the Eigen values of A, form the matrix A - λI­₂, find its determinant |A - λI₂| and solve the equation |A - λI₂| = 0:

Now,

|A  -λI₂| = (-3-λ)(1-λ)-(2)(-2) = λ²+2λ+1,

so that the eigenvalues of A are the roots of λ² + 2λ + 1 = (λ+1)², i.e. A has a repeated eigenvalue: λ₀ = -1.

Now we have to solve the system Au = λ₀u. Here λ₀ = -1, so that

Au = λ₀u ⇔ Au = -u,

which yields

or, equivalently,

A vector  is therefore an eigenvector associated with the eigen value λ₀ = -1 if and only if its coordinates satisfy

,

i.e. if and only if x₁ = x₂.

Hence the eigenvectors associated with the eigen value λ₀ = -1 are of the form , where α is a real number.

Diagonalisation of Matrix

The elements of a square matrix which lie in the sane row and same column position are said to form the diagonal of a square matrix. i.e. in a square matrix  the elements  are said to form the diagonal of a matrix .

For example in the matrix  the elements 1, 0 and 4 are the (1,1), (2,2) and (3,3) elements and hence form the diagonal of the square matrix A.

A square matrix is said to be a diagonal matrix if its non-diagonal elements are all zeros, where diagonal elements may or may not zero.

i.e. If  is a square matrix of order ‘n’ then it will be a diagonal matrix if a­_ij = 0, ∀ i ≠ j, where 1 ≤ i, j ≤ n.