Diagonalization Checker
Enter a square matrix, choose the working field, and test whether the matrix has enough independent eigenvectors for diagonalization.
Formula Used
A square matrix A is diagonalizable when there exists an invertible matrix P and a diagonal matrix
D such that A = P D P-1.
The checker uses these ideas:
det(A - λI) = 0gives eigenvalues.- The eigenspace for λ is
ker(A - λI). - Geometric multiplicity is
dim ker(A - λI). - Algebraic multiplicity counts how many times λ appears.
- A matrix is diagonalizable when the total geometric multiplicity equals
n.
Over the real numbers, every required eigenvector must also be real. A matrix with non-real eigenvalues may still be diagonalizable over the complex numbers.
How to Use This Calculator
- Select the matrix size.
- Choose whether to test over real or complex numbers.
- Enter all matrix entries.
- Adjust tolerance if values are very small or nearly repeated.
- Click Check Diagonalization.
- Review eigenvalues, multiplicities, eigenspace bases, and the final decision.
- Download the result as CSV or PDF when needed.
Example Data Table
| Example | Matrix | Field | Expected Result | Reason |
|---|---|---|---|---|
| Diagonalizable 3 × 3 | [[4, 1, 0], [1, 4, 0], [0, 0, 2]] | Real | Yes | Symmetric matrices have a full eigenvector basis. |
| Defective 3 × 3 | [[4, 1, 0], [0, 4, 0], [0, 0, 2]] | Real | No | The repeated eigenvalue 4 lacks enough independent eigenvectors. |
| Rotation 2 × 2 | [[0, -1], [1, 0]] | Real / Complex | Real: No, Complex: Yes | Its eigenvalues are ±i, so real diagonalization fails. |
FAQs
1. What does diagonalizable mean?
A matrix is diagonalizable when it is similar to a diagonal matrix. That means it has enough linearly independent eigenvectors to build a basis.
2. Can repeated eigenvalues still allow diagonalization?
Yes. Repeated eigenvalues are acceptable when their eigenspaces are large enough. The geometric multiplicity must match the algebraic multiplicity for every repeated eigenvalue.
3. Why might a matrix fail this test?
Failure usually happens because one or more eigenvalues do not produce enough independent eigenvectors. Then the matrix is defective and cannot be diagonalized over that field.
4. What is the difference between real and complex checking?
Real checking allows only real eigenvectors and real eigenvalues. Complex checking allows complex eigenvalues and often diagonalizes matrices that fail over the real numbers.
5. Why is tolerance included?
Numerical algorithms produce tiny rounding noise. Tolerance groups nearly equal eigenvalues and treats tiny residual values as zero when estimating eigenspace dimensions.
6. Does a symmetric matrix always diagonalize?
Over the real numbers, every real symmetric matrix is diagonalizable and even orthogonally diagonalizable. That makes symmetric examples especially stable in computation.
7. Why are eigenvectors useful after diagonalization?
They simplify matrix powers, differential equations, recurrences, and matrix functions. Once diagonalized, many calculations reduce to easy operations on diagonal entries.
8. Can this calculator handle decimal entries?
Yes. You can enter integers or decimals. The checker uses numerical linear algebra, so slight approximation is normal when entries or eigenvalues are not exact.