Part 1 [Engineering Design Optimization]: What are these eigenvalues and eigenvectors?
What you will read?
A. The basic definition of eigenvalues and eigenvectors terms.
B. How to calculate and interpret them?
A. The basic definition of eigenvalues and eigenvectors terms
Eigenvalues and eigenvectors are important elements that are commonly used in any area that uses matrices.
They are extremely popular since they provide significant information about the matrix that we work on. And both of them are correlated with each other.
The eigenvector v of a matrix A is a special vector such that premultiplication by A only scales v by a scalar amount lambda (eigenvalue). That is a matrix transformation produces:
Av = (lambda)v
In this case, if the A is an nxn matrix, then A may have up to n linearly independent eigenvectors where each eigenvector has a corresponding eigenvalue lambda.
An eigenvalue can be positive, zero, negative or imaginary (situation 4), or even repeated.
B. How to calculate and interpret them?
1. How to find eigenvalues?
Finding the eigenvalues means finding the root of rows in a matrix when you make it equal to 0.
Let’s take A matrix as:
Step 1: Form the characteristic polynomial:
det( (lambda)I — A)
Apply this formula to matrix A.
Step 2: Solve the roots of the polynomial that we obtain from step 1.
Based on the equation at the end, it can be said that we have 2 real negative lambda values; -2, and -1.
2. How to find eigenvectors?
We will use the knowledge that we got about the eigenvalues above. And for each eigenvalue, we can find an eigenvector with the following method.
We know the equation which is: Av = (lambda)v
Then, v can be obtained from this equation by following operations:
Since we know 2 eigenvalues (lambdas as notation), we can obtain v for both of them. Let’s calculate when lambda = -1:
We are not able to obtain a direct number. Because the eigenvector is any 2-element column vector in which the two elements have equal magnitude and opposite signs.
where k can take any value as magnitude.
The other eigenvector can be calculated in the same way too, I hope you can give it a try as a practice and this tutorial can help you to understand this calculation structure.
3. Insights
Here are some basic insights:
- If A is nxn matrix it can have at most n distinct eigenvalues.
- If A has m distant eigenvalues, then there will be m linearly independent eigenvectors.
- If A has m = n distinct eigenvalues, then A is diagonalizable*.
https://medium.com/linear-algebra/part-24-diagonalization-and-similarity-of-matrices-fe9517af1fe5
