Eigenvalue Calculator
Find Eigenvalues ($\lambda$) and Eigenvectors ($\mathbf{v}$) with Visualization
$$ A\mathbf{v} = \lambda\mathbf{v} $$
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Eigenvalues & Eigenvectors
Geometric Visualization ($A\mathbf{v} \parallel \mathbf{v}$)
Step-by-Step Solution
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By Prof. David Anderson
Professor of Mathematics | 20+ Years Teaching Experience
“The ‘Eigen-problem’ ($A\mathbf{x} = \lambda \mathbf{x}$) is arguably the most important equation in all of applied mathematics. It explains everything from quantum states to how Google ranks websites. In my years of teaching, I’ve noticed students get stuck on the Characteristic Polynomial. I built this Eigenvalue Calculator to not just solve the math, but to visualize the geometry.”
The Ultimate Guide to Eigenvalues and Eigenvectors: Theory, Calculation, and Visuals
Mastering the Characteristic Equation ($\det(A – \lambda I) = 0$), Eigenspaces, and Diagonalization
⚡ Key Takeaways for Students
- An Eigenvector ($\mathbf{v}$) is a non-zero vector that changes only by a scalar factor when a linear transformation is applied.
- The Eigenvalue ($\lambda$) is that scalar factor. Positive $\lambda$ stretches; negative $\lambda$ reverses direction.
- Use our Eigenvalue Calculator to solve the Characteristic Polynomial $\det(A – \lambda I) = 0$.
- Finding eigenvectors is equivalent to finding the Null Space of $(A – \lambda I)$.
Welcome to the professor’s guide on Eigenvalues and Eigenvectors. These concepts are the heartbeat of Linear Algebra. They allow us to simplify complex matrix operations by finding the “natural axes” of a system.
The word “Eigen” comes from German, meaning “own” or “characteristic”. An eigenvector is a vector that keeps its own direction. Our free Eigenvalue Calculator above automates the tedious algebra of finding roots for 2×2 and 3×3 matrices, and visualizes the transformation in real-time.
1. The Core Equation: What is an Eigenvalue?
Definition: The Eigen-Equation
For a square matrix $A$, a scalar $\lambda$ is an Eigenvalue if there exists a non-zero vector $\mathbf{v}$ such that:
$$ A\mathbf{v} = \lambda \mathbf{v} $$
Here, $\mathbf{v}$ is the Eigenvector. The matrix $A$ acts on $\mathbf{v}$ just like a simple number $\lambda$ would—it stretches or shrinks it, but does not rotate it off its span.
2. The 3-Step Algorithm to Find Eigenvalues
To calculate eigenvalues by hand (or verify our calculator), follow this standard procedure.
Step 1: The Characteristic Equation
Algorithm
Rearrange $A\mathbf{v} = \lambda \mathbf{v}$ to $(A – \lambda I)\mathbf{v} = \mathbf{0}$. For a non-zero solution $\mathbf{v}$ to exist, the matrix $(A – \lambda I)$ must be singular (non-invertible). Therefore, its determinant must be zero:
$$ \det(A – \lambda I) = 0 $$
This determinant expands into a polynomial in $\lambda$ called the Characteristic Polynomial.
Step 2: Find Roots (The Eigenvalues)
Algebra
Solve the polynomial for $\lambda$.
• For a 2×2 matrix, it’s a quadratic equation ($\lambda^2 + \dots$).
• For a 3×3 matrix, it’s a cubic equation ($\lambda^3 + \dots$).
The roots are your eigenvalues. Note that they can be real, repeated, or complex numbers.
Step 3: Find Eigenvectors (Null Space)
Linear Algebra
For each $\lambda$, substitute it back into $(A – \lambda I)\mathbf{v} = \mathbf{0}$. Solve this homogeneous system (find the Null Space) to get the corresponding eigenvectors.
3. Detailed Example: A 2×2 Matrix
Let’s find the eigenvalues and eigenvectors of $A = \begin{bmatrix} 4 & 1 \\ 2 & 3 \end{bmatrix}$.
1. Characteristic Polynomial:
$$ \det \begin{bmatrix} 4-\lambda & 1 \\ 2 & 3-\lambda \end{bmatrix} = (4-\lambda)(3-\lambda) – (1)(2) $$
$$ = \lambda^2 – 7\lambda + 12 – 2 = \lambda^2 – 7\lambda + 10 $$
2. Solve for Roots:
$$ \lambda^2 – 7\lambda + 10 = (\lambda – 5)(\lambda – 2) = 0 $$
So, eigenvalues are $\lambda_1 = 5$ and $\lambda_2 = 2$.
3. Find Eigenvectors:
For $\lambda_1 = 5$: Solve $(A – 5I)\mathbf{v} = \mathbf{0}$.
$$ \begin{bmatrix} -1 & 1 \\ 2 & -2 \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \end{bmatrix} \implies -x + y = 0 \implies x = y $$
Eigenvector $\mathbf{v}_1 = \begin{bmatrix} 1 \\ 1 \end{bmatrix}$.
4. Professor’s Shortcut: Trace and Determinant
Before you start a long calculation, use this trick to check your work. For any $n \times n$ matrix:
$$ \sum \lambda_i = \text{Trace}(A) \quad (\text{Sum of diagonals}) $$
$$ \prod \lambda_i = \det(A) \quad (\text{Determinant}) $$
Check our example: Trace = $4+3=7$. Sum of roots = $5+2=7$. Determinant = $12-2=10$. Product of roots = $5 \times 2 = 10$. It matches!
5. Algebraic vs. Geometric Multiplicity
This is where students often lose points.
| Concept | Definition | Implication |
|---|---|---|
| Algebraic Multiplicity | How many times a root $\lambda$ appears in the characteristic polynomial. | Max possible eigenvectors. |
| Geometric Multiplicity | The dimension of the Eigenspace (Nullity of $A-\lambda I$). | Actual number of independent eigenvectors. |
| Defective Matrix | Geometric < Algebraic | Matrix cannot be diagonalized. |
6. Frequently Asked Questions (FAQ)
What if the eigenvalues are complex numbers?
If the characteristic polynomial has no real roots (e.g., a rotation matrix), the eigenvalues are complex numbers (with $i$). This means no real vector stays parallel; the transformation involves rotation. Our calculator handles real eigenvalues for visualization.
Why do we calculate Eigenvalues?
Applications are endless: Google PageRank uses the principal eigenvector of the web graph. Structural Engineering uses them to find natural frequencies (resonance). PCA (Data Science) uses them to reduce dimensionality.
Can a 3×3 matrix have 0 as an eigenvalue?
Yes. If $\lambda = 0$ is an eigenvalue, it means $\det(A) = 0$. The matrix is singular (non-invertible), and it has a non-trivial Null Space.
References & Further Reading
- Strang, G. (2016). Introduction to Linear Algebra (5th ed.). Wellesley-Cambridge Press. (Chapter 6: Eigenvalues and Eigenvectors).
- Lay, D. C. (2021). Linear Algebra and Its Applications (6th ed.). Pearson. (Section 5.1-5.3).
- 3Blue1Brown. “Eigenvectors and eigenvalues | Essence of linear algebra.” YouTube
Visualize the Transformation
Don’t just solve for x. Use our free Eigenvalue Calculator to see how the matrix stretches space, find the characteristic polynomial steps, and verify your 3×3 homework answers instantly.
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