Eigen values and eigen vectors with example

 Eigenvalues and eigenvectors are important concepts in linear algebra that are used to describe the behavior of linear transformations.

An eigenvalue of a matrix A is a scalar value λ that satisfies the equation |A - λI| = 0, where I is the identity matrix. The eigenvalues of a matrix are also known as the roots of the matrix's characteristic polynomial.

An eigenvector of a matrix A is a non-zero vector v that satisfies the equation Av = λv, where λ is an eigenvalue of the matrix. The eigenvectors of a matrix are also known as the "principal" or "proper" vectors of the matrix.

Eigenvalues and eigenvectors have a number of important properties and applications. For example, the eigenvalues of a matrix A can be used to find the matrix's trace (the sum of its diagonal elements) and determinant (a scalar value that describes the matrix's magnitude and orientation). The eigenvectors of a matrix can be used to diagonalize the matrix (transform it into a diagonal matrix), which can simplify certain calculations

Here is an example of finding the eigenvalues of a 2x2 matrix:

Consider the following matrix:

[2 3] [1 2]

To find the eigenvalues of this matrix, we need to find the values of λ that satisfy the equation |A - λI| = 0. The matrix A is:

[2 3] [1 2]

And the identity matrix I is:

[1 0] [0 1]

Substituting these into the equation, we get:

|[2 3] - λ[1 0]| = 0 |[1 2] [0 1]|

This simplifies to:

(2 - λ)(2 - λ) - (3)(1) = 0

Which gives us the characteristic polynomial:

λ^2 - 4λ + 1 = 0

To find the eigenvalues, we solve this equation for λ:

λ = (4 ± √(4^2 - 4(1))) / 2 = (4 ± √(16 - 4)) / 2 = (4 ± √(12)) / 2 = {2, 3}

Therefore, the eigenvalues of the matrix are 2 and 3.

Here is an example of finding the eigenvectors of a 2x2 matrix:

Consider the following matrix:

[2 3] [1 2]

We have already found that the eigenvalues of this matrix are 2 and 3. To find the eigenvectors, we need to find the non-zero vectors v that satisfy the equation Av = λv, where λ is an eigenvalue of the matrix.

For the eigenvalue λ = 2, the equation is:

[2 3] * v = 2 * v [1 2] [1 2]

This simplifies to:

[4 6] = [2 4] [1 2] [1 2]

The only solution to this equation is v = [1 1], which is an eigenvector of the matrix with eigenvalue 2.

For the eigenvalue λ = 3, the equation is:

[2 3] * v = 3 * v [1 2] [1 2]

This simplifies to:

[6 9] = [3 6] [1 2] [1 2]

The only solution to this equation is v = [1 1], which is also an eigenvector of the matrix with eigenvalue 3.

Therefore, the eigenvectors of the matrix are [1 1] with eigenvalue 2, and [1 1] with eigenvalue 3