Calculate the eigenvalues and eigenvectors for a square matrix (2x2, 3x3, or 4x4). Input your matrix entries and view the detailed results below.
Characteristic Polynomial p(λ):
0
Eigenvalues (λ):
λ1: 0
Algebraic Multiplicity: 1
Geometric Multiplicity: 1
λ2: 0
λ3: 0
Algebraic Multiplicity: 0
Geometric Multiplicity: 0
λ4: 0
Eigenvectors (v):
v1 (for λ1): 0
Verification: A * v1 ≈ λ1 * v1 (0)
v2 (for λ2): 0
Verification: A * v2 ≈ λ2 * v2 (0)
v3 (for λ3): 0
Verification: A * v3 ≈ λ3 * v3 (0)
v4 (for λ4): 0
Verification: A * v4 ≈ λ4 * v4 (0)
Interpretation:
The algebraic multiplicity of an eigenvalue is its multiplicity as a root of the characteristic polynomial. The geometric multiplicity is the dimension of its corresponding eigenspace (number of linearly independent eigenvectors).
The matrix is diagonalizable if and only if for every eigenvalue, its algebraic multiplicity equals its geometric multiplicity. Based on the calculations, the matrix is 0.
Eigenvalues and eigenvectors are fundamental concepts in linear algebra, revealing how a linear transformation stretches or shrinks vectors.
The characteristic polynomial is given by: characteristic_polynomial = det(A - λI)
characteristic_polynomial = det(A - λI)
Let's find the eigenvalues and eigenvectors for matrix A = [[2,1],[1,2]].
Understanding these values helps in analyzing matrix transformations, diagonalizability, and system stability.
Eigenvalues are scalar values that represent the factor by which eigenvectors are scaled during a linear transformation.
Eigenvectors are non-zero vectors that only change by a scalar factor when a linear transformation is applied.
They are crucial in fields like physics, engineering, and computer science for analyzing systems, stability, and data reduction.
