JEE Champion's Guide to Characteristic Equation, Eigenvalues & Eigenvectors
Master the complete theory and problem-solving strategies for one of the most important topics in JEE Linear Algebra.
Why Eigenvalues & Eigenvectors Matter in JEE
Based on analysis of JEE papers from 2008-2024, Eigenvalues and Eigenvectors consistently appear in 2-3 questions per paper, making them crucial for scoring. Mastering these concepts will help you:
- Solve matrix power problems quickly using diagonalization
- Understand linear transformations geometrically
- Tackle system of differential equations in JEE Advanced
- Solve characteristic equation problems efficiently
- Gain 4-8 marks in every JEE paper
Characteristic Equation & Eigenvalues
For a square matrix $A$, the characteristic equation is given by $|A - \lambda I| = 0$, where $\lambda$ represents eigenvalues.
Key Formula:
Characteristic Polynomial: $P(\lambda) = |A - \lambda I|$
Characteristic Equation: $P(\lambda) = 0$
Example: Find eigenvalues of $A = \begin{bmatrix} 2 & 1 \\ 1 & 2 \end{bmatrix}$
Step 1: Set up $|A - \lambda I| = 0$:
$\begin{vmatrix} 2-\lambda & 1 \\ 1 & 2-\lambda \end{vmatrix} = 0$
Step 2: Expand determinant: $(2-\lambda)^2 - 1 = 0$
Step 3: Simplify: $\lambda^2 - 4\lambda + 3 = 0$
Step 4: Solve: $(\lambda - 1)(\lambda - 3) = 0$
Step 5: Eigenvalues: $\lambda = 1, 3$
Finding Eigenvectors
For each eigenvalue $\lambda$, solve $(A - \lambda I)\mathbf{x} = \mathbf{0}$ to find corresponding eigenvectors.
Example: Find eigenvectors for previous matrix
For $\lambda = 1$:
Solve $(A - I)\mathbf{x} = 0$: $\begin{bmatrix} 1 & 1 \\ 1 & 1 \end{bmatrix}\begin{bmatrix} x_1 \\ x_2 \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \end{bmatrix}$
Equation: $x_1 + x_2 = 0 \Rightarrow x_1 = -x_2$
Eigenvector: $\mathbf{v}_1 = \begin{bmatrix} 1 \\ -1 \end{bmatrix}$ or any scalar multiple
For $\lambda = 3$:
Solve $(A - 3I)\mathbf{x} = 0$: $\begin{bmatrix} -1 & 1 \\ 1 & -1 \end{bmatrix}\begin{bmatrix} x_1 \\ x_2 \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \end{bmatrix}$
Equation: $-x_1 + x_2 = 0 \Rightarrow x_1 = x_2$
Eigenvector: $\mathbf{v}_2 = \begin{bmatrix} 1 \\ 1 \end{bmatrix}$ or any scalar multiple
Properties & Theorems (JEE Advanced Focus)
Essential properties that frequently appear in JEE Advanced problems.
Key Properties:
- Sum of eigenvalues = Trace of matrix
- Product of eigenvalues = Determinant of matrix
- Eigenvalues of $A^k$ are $\lambda^k$ for same eigenvectors
- If $A$ is invertible, eigenvalues of $A^{-1}$ are $1/\lambda$
- Eigenvalues of symmetric matrices are real
Example (JEE Advanced 2021):
If $A$ is a $3 \times 3$ matrix with eigenvalues $1, 2, 3$, find eigenvalues of $A^2 - 2A + I$
Step 1: For each eigenvalue $\lambda$ of $A$, eigenvalue of $f(A)$ is $f(\lambda)$
Step 2: $f(\lambda) = \lambda^2 - 2\lambda + 1 = (\lambda - 1)^2$
Step 3: Apply to each eigenvalue:
• For $\lambda = 1$: $(1-1)^2 = 0$
• For $\lambda = 2$: $(2-1)^2 = 1$
• For $\lambda = 3$: $(3-1)^2 = 4$
Step 4: Eigenvalues of $A^2 - 2A + I$ are $0, 1, 4$
🚀 JEE Problem-Solving Strategies
For 2×2 Matrices:
- Use shortcut: $\lambda^2 - (\text{trace})\lambda + \text{det} = 0$
- Check if matrix is symmetric for real eigenvalues
- For triangular matrices, eigenvalues = diagonal entries
- Use properties to verify your answers
For 3×3 Matrices:
- Look for one obvious eigenvalue first
- Use factor theorem to simplify characteristic polynomial
- Check if matrix has special properties
- Use trace and determinant for verification
Core Concepts 4-8 Available in Full Version
Includes Diagonalization, Cayley-Hamilton Theorem, Similar Matrices, and Applications with JEE-level problems
📝 JEE Level Practice Problems
Test your understanding with these typical JEE problems:
1. Find eigenvalues and eigenvectors of $\begin{bmatrix} 3 & 1 \\ 1 & 3 \end{bmatrix}$
2. If $A$ has eigenvalues $2, 3, -1$, find determinant of $A^3 - 2A$
3. Prove that similar matrices have same eigenvalues
4. Find characteristic equation of $\begin{bmatrix} 1 & 2 & 0 \\ 2 & -1 & 0 \\ 0 & 0 & 3 \end{bmatrix}$
🎯 Must-Know Theorems for JEE Advanced
Cayley-Hamilton Theorem
Every square matrix satisfies its own characteristic equation: $P(A) = 0$
Spectral Theorem
A symmetric matrix can be diagonalized by an orthogonal matrix
Perron-Frobenius Theorem
Positive matrices have unique largest positive eigenvalue
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