Part 22 : Eigenvalues and Eigenvectors

Before learning about eigenvectors and eigenvalues we have to know that

when we multiply a scalar (like 2) with a vector (like vector b), we get a new vector and it points in the same direction as the original vector.

Magnitude of vector c is twice the magnitude of vector b, but they both point in the same direction

Now, assuming a square matrix A of order 3x3

Matrix A

If we multiply a vector x with matrix A, we get another vector in the product (multiplied with a scalar). If that product vector points in the same direction as vector x then it will be called an eigenvector of matrix A.

Lambda represents a scalar value

and product scalar will be called an eigenvalue of matrix A.

There could be multiple eigenvalues and eigenvectors for a symmetric and square matrix.

Finding eigenvectors and eigenvalues

We can manipulate the equation above to obtain characteristic equation.

Here I is an identity matrix of same order as matrix A

If vector x is not a zero vector then

Characteristic equation of matrix A

If we expand the determinant we will get an equation in terms of lambda and the roots of that equation will be eigenvalues of matrix A.

Corresponding eigenvectors could be calculated easily by substituting the eigenvalues.

Example

Say we have to find eigenvalues and eigenvectors of matrix G.

Matrix G

First we will obtain characteristic equation from matrix G

then we expand determinant to form an equation in terms of lambda.

Notice that this is a quadratic equation. The method of solving equation might vary depending on the degree.

Finally we will find the values of lambda (eigenvalues) by solving the equation.

We have the eigenvalues now we have to find eigenvectors. Starting with lambda = 5

After performing matrix multiplication we get

and we can conclude that

The ratio of x11 to x12 is 1 : (-1) so the first eigenvector of matrix G is

Similarly we can find eigenvector of matrix G when lambda = (-1)

and the second eigenvector of matrix G is

Facts

1. Sum of diagonal elements of any matrix is called trace. Sum of eigenvalues is equal to trace.

For matrix A

2. Product of eigenvalues of any square matrix is equal to determinant of that matrix.

For matrix A

3. If the eigenvalue is 0 then eigenvector lies in null space (eigenvector could not be a zero vector).

4. If matrix is squared (by matrix multiplication with itself) then the eigenvectors stay same but the eigenvalues are squared.

Read Part 23 : Orthonormal Vectors, Orthogonal Matrices and Hadamard Matrix

You can view the complete series here
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