Matrix Diagonalization Calculator
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The online Matrix Diagonalization Calculator helps you calculate instantly and solve problems related to Matrices Determinants. This tool provides accurate results using standard formulas and step-by-step calculation; you can view the formula with example in the calculator where available. Whether you are a student, teacher, or professional, this calculator simplifies complex calculations and saves time. Enter the required values below and get instant results. Results are shown clearly, with optional step-by-step explanation where applicable. The tool is free to use and works in any modern browser—no download or installation required. Bookmark this page for quick access whenever you need reliable Engineering Math calculations.
Diagonalize a 2×2 matrix A as A = P D P⁻¹ when a full eigenbasis exists. Distinguishes diagonalizable matrices (including repeated eigenvalues with geo=2) from defective matrices (geo < alg).
A = P D P⁻¹ (when A has a basis of eigenvectors)
A is diagonalizable iff it has n linearly independent eigenvectors. Repeated eigenvalues are still diagonalizable when geometric multiplicity equals algebraic multiplicity.
- Input is a real 2×2 matrix.
- This page reports real diagonalization A = P D P⁻¹.
- Eigenvectors are unique only up to nonzero scale; P may differ by column scaling/sign.
Inputs
Results
Worked Examples
Distinct eigenvalues (diagonalizable)
[[4,2],[1,3]] → λ=5,2 with independent eigenvectors; A = P D P⁻¹.
- matrix: [ [ 4, 2 ], [ 1, 3 ] ]
- result: Diagonalizable: YES
Repeated eigenvalue but diagonalizable
[[3,0],[0,3]] → λ=3 with alg=geo=2; still diagonalizable.
- matrix: [ [ 3, 0 ], [ 0, 3 ] ]
- result: Diagonalizable: YES
Defective / not diagonalizable
[[1,1],[0,1]] → λ=1 with alg=2, geo=1; NOT diagonalizable.
- matrix: [ [ 1, 1 ], [ 0, 1 ] ]
- result: Diagonalizable: NO
About this calculator
Overview
The Matrix Diagonalization Calculator decides whether a 2×2 matrix A is diagonalizable and, when it is, constructs P and D so that A = P D P⁻¹. Columns of P are eigenvectors; the diagonal of D holds the matching eigenvalues in the same order.
When to use
Use this for modal decoupling, checking hand diagonalizations, and distinguishing repeated-eigenvalue cases that are still diagonalizable from defective Jordan-block matrices.
Inputs explained
- Matrix A — real 2×2 matrix
Formula / method
Compute the characteristic polynomial and eigenvalues. For each eigenvalue find a basis of the eigenspace (nullspace of A−λI). If two independent eigenvectors exist, form P, D, compute P⁻¹, and verify both AP=PD and A=PDP⁻¹. If geometric multiplicity is too small, report NOT DIAGONALIZABLE.
Worked example
For A=[[4,2],[1,3]], eigenvalues are 5 and 2 with independent eigenvectors; A is diagonalizable. For A=[[1,1],[0,1]], λ=1 has algebraic multiplicity 2 but geometric multiplicity 1 — defective, not diagonalizable.
Interpreting results
Diagonalizable means A is similar to a diagonal matrix. A repeated eigenvalue does not by itself mean defective: [[3,0],[0,3]] is diagonalizable while [[1,1],[0,1]] is not.
Assumptions
- Real 2×2 input.
- Real diagonalization reported on this page.
- Eigenvector scaling/sign may vary; reconstruction must still recover A.
Limitations
- 2×2 only on this calculator.
- Complex eigenvalues are reported as not real-diagonalizable here.
- Educational tool — verify critical designs independently.
Important note
Educational engineering-mathematics tool. Verify critical designs with hand analysis or certified software.
How to Use This Calculator
- Enter the required values in the input fields.
- Click the Calculate button.
- View the computed result instantly.
Formula Used
Compute the characteristic polynomial and eigenvalues. For each eigenvalue find a basis of the eigenspace (nullspace of A−λI). If two independent eigenvectors exist, form P, D, compute P⁻¹, and verify both AP=PD and A=PDP⁻¹. If geometric multiplicity is too small, report NOT DIAGONALIZABLE.
Example Calculation
For A=[[4,2],[1,3]], eigenvalues are 5 and 2 with independent eigenvectors; A is diagonalizable. For A=[[1,1],[0,1]], λ=1 has algebraic multiplicity 2 but geometric multiplicity 1 — defective, not diagonalizable.
Frequently Asked Questions
What is Matrix Diagonalization Calculator?
- The Matrix Diagonalization Calculator decides whether a 2×2 matrix A is diagonalizable and, when it is, constructs P and D so that A = P D P⁻¹. Columns of P are eigenvectors; the diagonal of D holds the matching eigenvalues in the same order.
How does Matrix Diagonalization Calculator work?
- Compute the characteristic polynomial and eigenvalues. For each eigenvalue find a basis of the eigenspace (nullspace of A−λI). If two independent eigenvectors exist, form P, D, compute P⁻¹, and verify both AP=PD and A=PDP⁻¹. If geometric multiplicity is too small, report NOT DIAGONALIZABLE.
Why use this Engineering Math calculator?
- Use this for modal decoupling, checking hand diagonalizations, and distinguishing repeated-eigenvalue cases that are still diagonalizable from defective Jordan-block matrices.