Diagonalizable Matrix Test Calculator
Test whether square matrices are diagonalizable by analyzing eigenvalue multiplicities, checking linear independence of eigenvectors, and verifying diagonalization conditions. Get comprehensive analysis with geometric vs algebraic multiplicity comparisons.
Matrix Input
Diagonalizability Theory
What is Matrix Diagonalizability?
A square matrix A is diagonalizable if there exists an invertible matrix P such that P⁻¹AP = D, where D is a diagonal matrix. This is equivalent to saying that A has enough linearly independent eigenvectors to form a basis for the vector space.
Diagonalizability Condition:
A matrix A is diagonalizable if and only if:
Geometric Multiplicity = Algebraic Multiplicity
for every eigenvalue λ of A.
Where:
• Algebraic multiplicity = multiplicity of λ as a root of characteristic polynomial
• Geometric multiplicity = dimension of eigenspace E_λ = nullity(A - λI)
• Always: geometric multiplicity ≤ algebraic multiplicity
Diagonalizability Test Process
Step 1: Find Characteristic Polynomial
Compute det(A - λI) to get the characteristic polynomial
Step 2: Find All Eigenvalues
Solve characteristic equation to find eigenvalues and their algebraic multiplicities
Step 3: Find Eigenspaces
For each eigenvalue λ, find the eigenspace E_λ = null(A - λI)
Step 4: Calculate Geometric Multiplicities
Determine dim(E_λ) for each eigenvalue λ
Step 5: Compare Multiplicities
Check if geometric multiplicity = algebraic multiplicity for each λ
Step 6: Make Conclusion
Matrix is diagonalizable if and only if all multiplicities match
Special Cases and Properties
Distinct Eigenvalues
If A has n distinct eigenvalues, then A is diagonalizable
Each eigenspace has dimension 1
Symmetric Matrices
All real symmetric matrices are diagonalizable
Eigenvectors are orthogonal
Repeated Eigenvalues
Matrix may or may not be diagonalizable
Depends on geometric vs algebraic multiplicity
Triangular Matrices
Eigenvalues are diagonal entries
Diagonalizable if and only if it's diagonal
Applications of Diagonalizability
Matrix Powers
If A = PDP⁻¹, then Aⁿ = PDⁿP⁻¹ (easy to compute)
Differential Equations
Solving systems of linear ODEs with constant coefficients
Markov Chains
Long-term behavior analysis of stochastic processes
Principal Component Analysis
Data dimensionality reduction and feature extraction
Quantum Mechanics
Observable operators and state evolution
Vibration Analysis
Normal modes and natural frequencies in mechanical systems
Diagonalizability Test Examples
Example 1: Diagonalizable Matrix
Problem:
Test if the following matrix is diagonalizable:
[0 2]
Solution:
Step 1: Characteristic polynomial: det(A - λI) = (3-λ)(2-λ) = 0
Step 2: Eigenvalues: λ₁ = 3 (alg. mult. = 1), λ₂ = 2 (alg. mult. = 1)
Step 3: Eigenspaces: E₃ = span{[1; 0]}, E₂ = span{[1; -1]}
Step 4: Geometric multiplicities: dim(E₃) = 1, dim(E₂) = 1
Step 5: All geometric = algebraic multiplicities
Result: Matrix is DIAGONALIZABLE
Example 2: Non-Diagonalizable Matrix
Problem:
Test if the following matrix is diagonalizable:
[0 2]
Solution:
Step 1: Characteristic polynomial: det(A - λI) = (2-λ)² = 0
Step 2: Eigenvalue: λ = 2 (algebraic multiplicity = 2)
Step 3: Eigenspace: E₂ = null(A - 2I) = span{[1; 0]}
Step 4: Geometric multiplicity: dim(E₂) = 1
Step 5: Geometric multiplicity (1) ≠ algebraic multiplicity (2)
Result: Matrix is NOT DIAGONALIZABLE
Example 3: Symmetric Matrix
Problem:
Test if the following symmetric matrix is diagonalizable:
[2 1]
Solution:
Step 1: Characteristic polynomial: det(A - λI) = λ² - 2λ - 3 = 0
Step 2: Eigenvalues: λ₁ = 3, λ₂ = -1 (both alg. mult. = 1)
Step 3: Eigenspaces: E₃ = span{[1; 1]}, E₋₁ = span{[1; -1]}
Step 4: Geometric multiplicities: both equal to 1
Step 5: All geometric = algebraic multiplicities
Result: Matrix is DIAGONALIZABLE (as expected for symmetric matrices)
Frequently Asked Questions
Algebraic multiplicity is how many times an eigenvalue appears as a root of the characteristic polynomial. Geometric multiplicity is the dimension of the corresponding eigenspace. For diagonalizability, these must be equal for every eigenvalue.
Yes! If an n×n matrix has n distinct eigenvalues, it is always diagonalizable. This is because each eigenspace has dimension at least 1, and with n distinct eigenvalues, we get n linearly independent eigenvectors.
The Spectral Theorem guarantees that real symmetric matrices have real eigenvalues and orthogonal eigenvectors. The geometric multiplicity always equals the algebraic multiplicity for symmetric matrices, ensuring diagonalizability.
Non-diagonalizable matrices can still be put in Jordan canonical form, which is the closest we can get to diagonal form. Jordan form has eigenvalues on the diagonal and 1's on the superdiagonal in Jordan blocks.
The geometric multiplicity of eigenvalue λ is the dimension of the eigenspace E_λ = null(A - λI). This equals n - rank(A - λI), where n is the matrix size. You can find it by row-reducing A - λI and counting free variables.
Yes! The same diagonalizability conditions apply to complex matrices. In fact, over the complex numbers, every matrix has a complete set of eigenvalues (counting multiplicities), making diagonalizability analysis more straightforward.