Last updated: July 31, 2026
Eigenvalue and Eigenvector Calculator
Creators
Dharmendra SinghReviewers
Creators
Dharmendra SinghReviewers
Creators
Dharmendra SinghReviewers
Creators
Dharmendra SinghReviewers
Formula
For matrix [[a,b],[c,d]]: λ = (trace ± √(trace² - 4det))/2; eigenvectors solved from (A-λI)v=0
Where:
- a=a
- b=b
- c=c
- d=d
Worked Examples
Distinct real eigenvalues
Matrix [[4,1],[2,3]].
- 1trace=7, det=10
- 2λ=(7±3)/2 => 5 and 2
- 3Compute normalized eigenvectors
Diagonal matrix
Matrix [[2,0],[0,5]].
- 1Eigenvalues are diagonal entries
- 2Standard basis vectors are eigenvectors
Symmetric matrix
Matrix [[3,1],[1,3]].
- 1trace=6, det=8
- 2λ=4 and 2
- 3Compute orthogonal eigenvectors
Introduction
Find real eigenvalues and normalized eigenvectors for a 2×2 matrix, ideal for linear algebra checks and study.
Formula and Method
Eigenvalue and Eigenvector Calculator uses: For matrix [[a,b],[c,d]]: λ = (trace ± √(trace² - 4det))/2; eigenvectors solved from (A-λI)v=0
Enter all required inputs
Apply the underlying math formula(s)
Compute supporting terms where needed
Round numeric outputs for stable display
Input Fields
Provide values for: a, b, c, d.
- 1
a: a
- 2
b: b
- 3
c: c
- 4
d: d
Output Fields
The calculator returns: lambda1, lambda2, v1x, v1y, v2x, v2y.
- 1
lambda1: First real eigenvalue
- 2
lambda2: Second real eigenvalue
- 3
v1x: x-component of normalized eigenvector for lambda1
- 4
v1y: y-component of normalized eigenvector for lambda1
- 5
v2x: x-component of normalized eigenvector for lambda2
- 6
v2y: y-component of normalized eigenvector for lambda2
Validation Rules
Input validation is applied before calculation to prevent invalid math states.
All matrix entries must be finite numbers.
Only real eigenvalues are supported (discriminant must be non-negative).
Eigenvectors are normalized to unit length.
Outputs are rounded to 6 decimals.
How to Use This Calculator
Fill in the inputs, run the calculator, and review primary output first.
Enter the required values
Click calculate
Check the primary output
Use example scenarios for verification
Practical Uses
This calculator supports learning, quick checks, and day-to-day technical problem solving.
Homework and exam preparation
Engineering and science sanity checks
Classroom demonstrations
Fast recalculation with different inputs
FAQs
What does the Eigenvalue and Eigenvector Calculator compute?
Find real eigenvalues and normalized eigenvectors for a 2×2 matrix, ideal for linear algebra checks and study.
Which inputs are required?
Required inputs: a, b, c, d.
How should I format numbers?
Use finite numeric values; decimals and negative numbers are allowed unless restricted by validation rules.
How are results rounded?
Numeric outputs are rounded to about 6 decimal places for readability.
Why might I see an error?
All matrix entries must be finite numbers.
Can I use this for learning and verification?
Yes. Use the examples and formula section to verify manual calculations quickly.