Eigenvalues & Eigenvectors
Calculate the eigenvalues and corresponding eigenvectors of a matrix. Get instant step-by-step solutions — completely free, no signup required.
Free Eigenvalues & Eigenvectors Calculator
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About Eigenvalues & Eigenvectors
Calculate the eigenvalues and corresponding eigenvectors of a matrix. Linear algebra and matrix operations are foundational to computer graphics, machine learning, and solving systems of equations. Understanding matrix manipulations is critical for advanced mathematics.
Our Eigenvalues & Eigenvectors calculator handles every type of problem in this topic. Whether you are a student working through homework, preparing for an exam, or a professional needing a quick calculation, you can enter any eigenvalues & eigenvectors expression and receive a complete, verified solution with every step shown.
Eigenvalues & Eigenvectors Formula
det(A − λI) = 0, (A − λI)v = 0
An eigenvector of A is a non-zero vector that A only stretches, never rotates: Av = λv. The scale factor λ is its eigenvalue. Find the eigenvalues by solving the characteristic equation det(A − λI) = 0, then find each eigenvector by solving (A − λI)v = 0.
Worked Example
Find the eigenvalues and eigenvectors of A = [[4, 1], [2, 3]]
- Characteristic equation: (4 − λ)(3 − λ) − 2 = λ² − 7λ + 10 = 0
- Factor: (λ − 5)(λ − 2) = 0, so λ = 5 or λ = 2
- λ = 5: (A − 5I)v = 0 gives −x + y = 0 → v = (1, 1)
- λ = 2: (A − 2I)v = 0 gives 2x + y = 0 → v = (1, −2)
Answer: λ = 5 with (1, 1); λ = 2 with (1, −2)
Common Mistakes to Avoid
- Subtracting λ from every entry instead of only the diagonal.
- Reporting the zero vector as an eigenvector.
- Forgetting that any non-zero multiple of an eigenvector is also an eigenvector.
How to Use the Eigenvalues & Eigenvectors Calculator
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Input your matrix or vector values for the Eigenvalues & Eigenvectors problem
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Press Enter or click the Solve button to run the calculation
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Review the step-by-step solution that appears below the input
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Check the breakdown of each operation to understand the method
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Try a different problem to practice and build your understanding
Frequently Asked Questions
What is the Eigenvalues & Eigenvectors formula?
det(A − λI) = 0, (A − λI)v = 0. An eigenvector of A is a non-zero vector that A only stretches, never rotates: Av = λv. The scale factor λ is its eigenvalue. Find the eigenvalues by solving the characteristic equation det(A − λI) = 0, then find each eigenvector by solving (A − λI)v = 0.
Can you show a worked Eigenvalues & Eigenvectors example?
Find the eigenvalues and eigenvectors of A = [[4, 1], [2, 3]]: Characteristic equation: (4 − λ)(3 − λ) − 2 = λ² − 7λ + 10 = 0; Factor: (λ − 5)(λ − 2) = 0, so λ = 5 or λ = 2; λ = 5: (A − 5I)v = 0 gives −x + y = 0 → v = (1, 1); λ = 2: (A − 2I)v = 0 gives 2x + y = 0 → v = (1, −2). Answer: λ = 5 with (1, 1); λ = 2 with (1, −2).
Is this Eigenvalues & Eigenvectors calculator free to use?
Yes — the Eigenvalues & Eigenvectors calculator on MathSimplify is completely free. There is no signup, no subscription, and no usage limit. Enter any problem and get the step-by-step solution instantly.
Can I use this for Eigenvalues & Eigenvectors homework and exam practice?
Absolutely. The step-by-step breakdown is designed specifically for learning. You can use it to check your work, understand where you went wrong, or study the method before an exam. The calculator shows the full working, not just the final answer.