Diagonalize Matrix
Diagonalize any 2x2 matrix by finding its eigenvalues and eigenvectors. Get the diagonal matrix D = P⁻¹AP with step-by-step linear algebra.
About This Calculator
The 2x2 Matrix Diagonalization Calculator finds the eigenvalues and eigenvectors of any 2x2 matrix, constructs the diagonalization D = P⁻¹AP, and checks whether the matrix is diagonalizable over the real numbers. Simply enter the four elements of your 2x2 matrix and click Diagonalize.
A matrix is diagonalizable when it has a full set of linearly independent eigenvectors -- two for a 2x2 matrix. Distinct eigenvalues guarantee diagonalizability. The calculator checks for defective cases (repeated eigenvalues with only one eigenvector) and complex eigenvalues.
Frequently Asked Questions
What does it mean to diagonalize a matrix?
Diagonalizing a matrix A means finding an invertible matrix P and a diagonal matrix D such that D = P⁻¹AP.
When is a matrix diagonalizable?
A 2x2 matrix is diagonalizable over the real numbers if it has two linearly independent eigenvectors (real).
What are eigenvalues and eigenvectors?
Eigenvalues are scalars lambda such that Av = lambdav for some non-zero vector v (the eigenvector).
How is diagonalization used in practice?
Diagonalization is used to compute high powers of matrices, solve systems of differential equations, and perform PCA.
Is this tool free to use?
Yes, all calculators on Calculy are completely free to use with no registration required.