A square matrix is diagonalizable if and only if it possesses a complete set of linearly independent eigenvectors. This fundamental condition determines whether a matrix can be transformed into a simple diagonal form.
Understanding Diagonalization
Diagonalization is the process of transforming a square matrix (let's call it A) into a diagonal matrix (D) using a similarity transformation. This means expressing A in the form A = CDC⁻¹, where:
- D is a diagonal matrix whose diagonal entries are the eigenvalues of A.
- C is an invertible matrix whose columns are the corresponding linearly independent eigenvectors of A.
- C⁻¹ is the inverse of matrix C.
This transformation simplifies many matrix operations, such as computing powers of a matrix or solving systems of linear differential equations.
Key Conditions for Diagonalizability
For an n x n matrix A, the core condition for diagonalizability is that it must have n linearly independent eigenvectors. If this condition is met, the matrix is diagonalizable.
To determine if an n x n matrix A has n linearly independent eigenvectors, you typically check the relationship between its algebraic multiplicity (AM) and geometric multiplicity (GM) for each eigenvalue.
1. Algebraic Multiplicity (AM)
The algebraic multiplicity of an eigenvalue (λ) is the number of times it appears as a root of the characteristic polynomial, det(A - λI) = 0.
2. Geometric Multiplicity (GM)
The geometric multiplicity of an eigenvalue (λ) is the dimension of its eigenspace, which is the number of linearly independent eigenvectors associated with that eigenvalue. It can be found by calculating the dimension of the null space of the matrix (A - λI).
The Rule for Diagonalizability
A matrix A is diagonalizable if and only if two conditions are met:
- For every eigenvalue (λ) of A, its Geometric Multiplicity (GM) must equal its Algebraic Multiplicity (AM).
- The sum of the geometric multiplicities for all distinct eigenvalues must equal n (the dimension of the matrix).
If, for any eigenvalue, GM < AM, then the matrix is not diagonalizable. The geometric multiplicity can never be greater than the algebraic multiplicity (GM ≤ AM).
Comparing Multiplicities
| Condition for Eigenvalue λ | Result for Matrix A | Explanation |
|---|---|---|
| GM(λ) = AM(λ) | Potentially Diagonalizable | This condition must hold for all eigenvalues. |
| GM(λ) < AM(λ) | Not Diagonalizable | The matrix lacks a sufficient number of linearly independent eigenvectors for this specific eigenvalue. |
Practical Steps to Check Diagonalizability
To determine if an n x n matrix A is diagonalizable, follow these steps:
- Find the Eigenvalues: Calculate the eigenvalues by solving the characteristic equation
det(A - λI) = 0, where I is the identity matrix. - Determine Algebraic Multiplicity (AM): For each unique eigenvalue, count how many times it appears as a root of the characteristic polynomial.
- Determine Geometric Multiplicity (GM): For each unique eigenvalue (λ):
- Form the matrix
(A - λI). - Find the null space of
(A - λI)by solving the system(A - λI)v = 0. - The dimension of this null space is the geometric multiplicity. This can often be found by calculating
n - rank(A - λI).
- Form the matrix
- Compare AM and GM: For every distinct eigenvalue, verify if GM = AM.
- Check Total Eigenvectors: Ensure that the sum of the geometric multiplicities equals n.
If both steps 4 and 5 are true, the matrix is diagonalizable.
Special Cases: When Diagonalizability is Guaranteed
There are certain types of matrices that are always diagonalizable:
- Matrices with Distinct Eigenvalues: If an n x n matrix has n distinct eigenvalues, it is guaranteed to be diagonalizable. This is because each distinct eigenvalue will have an AM of 1, and its GM will also be 1, satisfying the condition GM = AM for all eigenvalues.
- Symmetric Matrices: All real symmetric matrices are diagonalizable. Furthermore, their eigenvectors corresponding to distinct eigenvalues are orthogonal.
Understanding these conditions allows you to efficiently determine whether a given matrix can be simplified through diagonalization, which is crucial for various applications in linear algebra and beyond.
