Matrix diagonalization checker. Enter A to find eigenvalues/vectors, test diagonalizability and output P and D—for similarity.
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P⁻¹ A P = D
space-separated entries, semicolon-separated rows, e.g. 1 2;3 4
Result
📖 Tutorial｜Matrix Diagonalization
1. Definition
A is diagonalizable if there exists invertible P with P⁻¹AP = D (D diagonal).
2. Symbols
Symbol
Meaning
P
matrix whose columns are eigenvectors
D
diagonal matrix of eigenvalues
n
size of A
3. How this tool works
A is diagonalizable iff it has n linearly independent eigenvectors;
Columns of P are the eigenvectors;
Diagonal of D follows the column order of P.
4. Steps
Enter A;
Click Calculate to see eigenvalues and whether A is diagonalizable.
5. Example
Example: A=2 1;1 2. Eigenvalues 3,1 each have an eigenvector; P=[(1,1),(−1,1)], D=diag(3,1).
6. Pitfalls
Complex eigenvalues are not diagonalizable over R; Repeated eigenvalues need enough independent eigenvectors; Column order of P must match diagonal of D.
❓ FAQ｜Matrix Diagonalization
When is A diagonalizable?
When it has n linearly independent eigenvectors.
Which matrices are always diagonalizable?
Real symmetric matrices are orthogonally diagonalizable.
How to build P and D?
Columns of P are eigenvectors; diagonal of D holds the corresponding eigenvalues.