Finding Eigenvalues (Step-by-Step)
To find the eigenvalues of a square matrix A, we use a special equation called the characteristic equation.
The characteristic equation is:
det(A - λI) = 0
Here:
- λ (lambda) represents the eigenvalue (we are trying to find it)
- I is the identity matrix (same size as A)
- det() means determinant of a matrix
🔍 Example:
Let’s say we have a 2x2 matrix:
A = [[4, 2], [1, 3]]
Step 1: Subtract λ from the diagonal entries of A:
A - λI = [[4−λ, 2], [1, 3−λ]]
Step 2: Find the determinant of this matrix and set it to 0:
det(A - λI) = (4−λ)(3−λ) − (2×1) = 0
=> λ² − 7λ + 10 = 0
Step 3: Solve this quadratic equation:
λ = 5 and λ = 2
✅ So, the eigenvalues are 5 and 2.
Finding Eigenvectors (Step-by-Step)
After finding eigenvalues, we find the corresponding eigenvectors using this equation:
(A − λI)x = 0
This is a system of linear equations. We solve it like we solve any system — using substitution,
elimination, or matrices. The goal is to find a non-zero vector x that satisfies
the equation.
🔍 Example (continued):
We already found the eigenvalues 5 and 2 for matrix A = [[4, 2], [1, 3]].
Let’s find the eigenvector for λ = 5.
Subtract 5 from the diagonal of A:
(A − 5I) = [[−1, 2], [1, −2]]
Solve: (A − 5I)x = 0 → [[-1, 2], [1, -2]] [x1, x2]ᵗ = [0, 0]
This gives: −x1 + 2x2 = 0 → x1 = 2x2
So, one possible eigenvector is:
x = [2, 1]ᵗ (you can scale it to any multiple)
💡 You’ll always get infinite possible eigenvectors for a given eigenvalue — because any scalar multiple is still an eigenvector!
Key Properties to Remember
- Sum of eigenvalues = Trace of the matrix (sum of diagonal elements)
- Product of eigenvalues = Determinant of the matrix
- If one eigenvalue is 0, the matrix is not invertible
- Eigenvectors with different eigenvalues are linearly independent
- If the matrix is symmetric, all eigenvalues are real
- Symmetric matrices can be diagonalized using orthogonal matrices
💡 Applications include:
- Solving systems of differential equations
- Principal Component Analysis (PCA) in machine learning
- Stability analysis in engineering
- Vibrations and quantum mechanics