Key takeaways
- λ = (trace ± √(trace² − 4 × det)) ÷ 2, where trace = a + d and det = ad − bc.
- The two eigenvalues add up to the trace and multiply to the determinant.
- A negative number under the square root means complex eigenvalues.
- The calculator's main result is the larger eigenvalue. Both are shown in the steps.
In this guide
What is an eigenvalue?
When a matrix acts on arrows (vectors), most arrows get turned to a new direction. But a few special arrows keep their direction and only get longer or shorter. These are eigenvectors, and the amount each one is stretched is its eigenvalue, written λ (lambda).
An eigenvalue of 3 means 'this direction is stretched 3 times'. An eigenvalue of 0.5 means it is shrunk to half.
Why do eigenvalues matter?
Eigenvalues reveal what a matrix does in the long run. Apply the same matrix again and again, and the direction with the biggest eigenvalue takes over. That is why they are used for population growth, vibrations and even ranking web pages.
For the matrix [5 2; 2 2], the arrow (2, 1) goes to (12, 6), which is exactly 6 times the original. So 6 is an eigenvalue and (2, 1) is its eigenvector.
The formula
The eigenvalues solve the characteristic equation λ² − (a + d)λ + (ad − bc) = 0. Using the quadratic formula:
| Symbol | What it means | Example |
|---|---|---|
| a, b, c, d | Entries of the matrix [a b; c d] | 5, 2, 2, 2 |
| a + d | The trace | 7 |
| ad − bc | The determinant | 6 |
| (a + d)² − 4(ad − bc) | The discriminant | 25 |
| λ₁, λ₂ | The two eigenvalues (+ and −) | 6 and 1 |
The calculator shows both eigenvalues in its steps and gives the larger one as the main result. If the discriminant is negative, it tells you the eigenvalues are complex and shows them in its steps in the form a ± bi.
How to find eigenvalues step by step
- 1Find the trace.
a + d. - 2Find the determinant.
ad − bc. - 3Find the discriminant.
trace² − 4 × determinant. - 4Take its square root. If it is negative, the eigenvalues are complex.
- 5Split into two.
λ₁ = (trace + root) ÷ 2andλ₂ = (trace − root) ÷ 2.
Now try your own matrix in the calculator below and compare both eigenvalues in the steps.
Try it yourself
Pre-filled with the example — change any value.
Common mistakes to avoid
- Forgetting the second eigenvalue. A 2×2 matrix has two (they may be equal). The main result shows only the larger one.
- Sign errors in ad − bc, especially with negative entries.
- Dividing only the square root by 2. The whole top, trace ± root, is divided by 2.
Tips and tricks
- Triangular matrix? If b = 0 or c = 0, the eigenvalues are simply a and d. For [2 0; 0 3] they are 2 and 3.
- Symmetric matrix (b = c)? Its eigenvalues are always real numbers.
- Rotation matrices like [0 −1; 1 0] have complex eigenvalues (here ±i), because no real direction stays fixed.
Where you'll use it in real life
- Engineering: checking whether a system settles down or blows up.
- Buildings and bridges: finding natural vibration frequencies.
- Population models: the largest eigenvalue gives the long-run growth rate.
- Differential equations: the same idea solves second-order equations.
Quick summary
To find the eigenvalues of a 2×2 matrix, work out the trace and determinant, then use λ = (trace ± √(trace² − 4 × det)) ÷ 2. Check that they add to the trace and multiply to the determinant. Use the calculator above to see both eigenvalues with every step shown.
Worked example
A population model uses the matrix [5 2; 2 2] to describe how two groups change from one year to the next. You want its eigenvalues to see how fast the population grows in the long run.
- 01λ = (5+2 ± √((5+2)²-4(5×2-2×2)))/2
- 02= (7 ± √(49-24))/2
- 03= (7 ± √25)/2
- 04= (7 ± 5)/2
- 05λ₁ = 6, λ₂ = 1
- 06Using λ = 6
These numbers are pre-filled in the calculator above.
Frequently asked questions
How do I check my eigenvalues?
Add them: the sum must equal the trace a + d. Multiply them: the product must equal the determinant ad − bc.
Can eigenvalues be negative or zero?
Yes. A negative eigenvalue flips that direction. An eigenvalue of 0 means the matrix squashes that direction flat, and the determinant is 0.
What if the discriminant is negative?
Then the eigenvalues are complex numbers. This usually means the matrix includes a rotation, so no real direction stays fixed.
Which eigenvalue does the calculator show as the result?
The larger of the two real eigenvalues. Both eigenvalues are listed in the step-by-step working.
