For AI agents: use /llms.txt for the Nakafa content index.
To find the eigenvalues of a matrix, we need a very important mathematical tool in linear algebra. Imagine we want to find all values that make the matrix become singular (not invertible).
Let . We can form a special function:
This function is a polynomial of degree in , which we call the characteristic polynomial of .
with coefficients .
In fact, is indeed a polynomial of degree for every matrix .
Let be a square matrix. The trace of is the sum of the diagonal elements:
The matrix trace has a close relationship with the coefficients of the characteristic polynomial.
In the characteristic polynomial of , its coefficients have special meaning:
This means:
The most important concept of the characteristic polynomial is its relationship with eigenvalues.
Now, let's look at a very important relationship: is an eigenvalue of if and only if .
In other words, eigenvalues are the roots of the characteristic polynomial.
The equation for :
is what we call the characteristic equation of .
Now, what happens if an eigenvalue appears multiple times as a root of the characteristic polynomial? Let and . The multiplicity of the root of the characteristic polynomial is called the algebraic multiplicity of the eigenvalue of . We say is an eigenvalue with multiplicity of .
For every eigenvalue , it holds:
One important result in eigenvalue theory is the relationship between geometric and algebraic multiplicity.
For any matrix and , we have an interesting relationship:
The geometric multiplicity of every eigenvalue is always less than or equal to its algebraic multiplicity.
Why does this happen? This can be explained using basis transformation and Jordan block form.
After understanding the basic concepts, let's see how to apply them in concrete examples of characteristic polynomial calculation:
Let . The characteristic polynomial of is:
For , only has the root with algebraic multiplicity .
For , has roots , , and with algebraic multiplicities respectively.
The characteristic polynomial of matrix is:
This matrix has the root with algebraic multiplicity . is the only eigenvalue of . We have calculated that .
Now, let's explore something fascinating: how the characteristic polynomial works on common geometric transformations we often encounter in :
Rotation with has the characteristic polynomial:
This has real roots if and only if , that is , so or .
Reflection along axes with has the characteristic polynomial:
The eigenvalues are and with .
Scaling with has the characteristic polynomial:
The eigenvalue is with .
Shearing with with has the characteristic polynomial:
The eigenvalue is with .
Similar matrices have a fascinating property: they have the same characteristic polynomial, and therefore have the same eigenvalues, the same trace, and the same determinant.
Let's see why this is true. Let be invertible and . Then:
Now, what about the eigenvectors of similar matrices? Let be similar matrices with and invertible matrix . If is an eigenvalue of both and , and is an eigenvector of for eigenvalue , then is an eigenvector of for eigenvalue .
Let's see why this is true. Let and . Then:
This shows that similarity transformation not only preserves eigenvalues, but also provides a systematic way to transform eigenvectors.
