The Eigenvector Calculator is a digital tool that calculates the eigenvectors of a square matrix. An eigenvector of a matrix is a non-zero vector that remains in the same direction when the matrix is applied as a transformation, although it may be scaled by a corresponding eigenvalue.
The tool automatically computes eigenvectors using the eigenvalues of the matrix, solving the equation:
(A − λI)v = 0
Where A is the square matrix, λ is an eigenvalue of A, v is the eigenvector associated with λ, and I is the identity matrix.
Eigenvectors:
Non-zero vectors that do not change direction under a matrix transformation, only scaled by the corresponding eigenvalue.
Eigenvalues:
Scalars that represent how much the eigenvector is stretched or compressed during the transformation.
Matrix:
A rectangular array of numbers arranged in rows and columns, representing a linear transformation.
Linear Algebra:
The mathematical field that studies vectors, matrices, and linear transformations, where eigenvectors and eigenvalues are crucial for solving problems.
The eigenvector is computed using the standard equation:
(A - λI)v = 0
Where:
A = square matrix
λ = eigenvalue of A
v = eigenvector corresponding to λ
I = identity matrix
Steps to find an eigenvector manually:
1. Find eigenvalues λ using: det(A − λI) = 0
2. Solve the system: (A − λI)v = 0
3. Determine the eigenvectors v up to a scalar multiple.
Pro Tip: Highlight these formulas in a frame for clear visibility and better user experience.Problem: Find the eigenvectors of the matrix:
A = [3 1]
[0 2]
Step 1:
Find eigenvalues using the determinant:
det(A − λI) = 0
|3 − λ 1|
|0 2 − λ| = 0
(3 − λ)(2 − λ) − (0 * 1) = 0
λ₁ = 3, λ₂ = 2
Step 2:
Solve (A − λI)v = 0 for each eigenvalue.
For λ₁ = 3:
(A − 3I)v = [0 1] [x] = 0
[0 -1] [y]
0x + 1y = 0 ⟹ y = 0
Eigenvector: v₁ = [1] (up to scalar multiple)
[0]
For λ₂ = 2:
(A − 2I)v = [1 1] [x] = 0
[0 0] [y]
x + y = 0 ⟹ y = −x
Eigenvector: v₂ = [ 1] (up to scalar multiple)
[-1]