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Chapter 4. Eigenvalues and Eigenforms

Recommended article : 【Linear Algebra】 Linear Algebra Contents

1. Eigenvalues and Eigenvectors

2. Diagonalization of Matrices

3. Quadratic Forms

4. Differential Equations and Eigenvalues

1. Eigenvalues and Eigenvectors

⑴ Eigenvalues and eigenvectors

① Used in diagonalizability, spectral clustering, etc.

② Theorem 1. The necessary and sufficient condition for A - λI to have a non-zero kernel in (A - λI)x = 0 is det(A - λI) = 0

○ Proof : rank-nullity theorem

③ Theorem 2. When the eigenvalues of A ∈ ℝ2×2 are λ1, λ2, A2 - (λ1 + λ2)A + λ1λ2I = O holds

○ Proof

From the definition of eigenvalues, the following holds

From this, it is easily known that λ1 + λ2 = a + d, and λ1λ2 = ad - bc

Therefore, by the [Cayley-Hamilton theorem](https://jb243.github.io/pages/1897#:1.-,% B4 ,-(CayleyHamiltontheorem), the proposition holds

Applying this to calculate An as (A - λ1I)(A - λ2I) Q(A) + kA + sE for An (for n ≥ 2)

④ Theorem 3. By applying the diagonalization of matrices, A = PDP-1, we can obtain An = PDnP-1

⑤ Theorem 4. For real matrices A and B of size n × n, the eigenvalues of matrix AB are the same as the eigenvalues of matrix BA (ref-,5,-.))

○ Proof : Assume λ is an eigenvalue of AB

⇔ ABv = λv, v ≠ 0

⇔ BABv = BA (Bv) = B · λv = λ (Bv)

⇔ Considering all possible cases as follows, the eigenvalues of BA are the same as those of AB

○ When Bv ≠ 0 : λ is an eigenvalue of BA

○ When Bv = 0

⇔ ABv = A0 = 0 = λv

⇔ λ = 0 is an eigenvalue of AB

⇔ 0 = det(AB) = det(A) × det(B) = det(BA)

⇔ Hence, there exists v’ ≠ 0 such that BAv’ = 0

⇔ λ = 0 is an eigenvalue of BA

⑥ Application 1. Hückel approximation and Hamiltonian

⑦ Python code (cf. WolframAlpha)

○ Using numpy

import numpy as np
matrix = np.array([
    [4.56, 10.2, 7.68, 13.68], 
    [10.2, 33.2, 20.2, 24.8],  
    [7.68, 20.2, 17.84, 23.44],
    [13.68, 24.8, 23.44, 45.55]
])
eigenvalues, eigenvectors = np.linalg.eigh(matrix)
sorted_indices = np.argsort(eigenvalues)[::-1]
sorted_eigenvalues = eigenvalues[sorted_indices]
sorted_eigenvectors = eigenvectors[:, sorted_indices]
print("Eigenvalues:\n", sorted_eigenvalues)
print("\nEigenvectors:\n", sorted_eigenvectors)

○ Using sympy

from sympy import Matrix

# Create SymPy matrix
matrix = Matrix([
    [4.56, 10.2, 7.68, 13.68], 
    [10.2, 33.2, 20.2, 24.8],  
    [7.68, 20.2, 17.84, 23.44],
    [13.68, 24.8, 23.44, 45.55]
])

# Calculate eigenvaleus and eigenvectors 
eigenvalues = matrix.eigenvals()  # 고유값 계산
eigenvectors = matrix.eigenvects()  # 고유벡터 계산

# Output 
print("Eigenvalues:")
for eigval, multiplicity in eigenvalues.items():
    print(f"Value: {eigval}, Multiplicity: {multiplicity}")

print("\nEigenvectors:")
for eigval, eigen_mult, eigvecs in eigenvectors:
    print(f"Eigenvalue: {eigval}")
    for vec in eigvecs:
        print(f"Eigenvector: {vec}")

⑵ Generalized eigenvectors

① Overview

○ Definition : When the geometric multiplicity of an asymmetric matrix is less than the algebraic multiplicity, vectors that replace the lacking eigenvectors

○ The generalized eigenvectors of matrix A of rank m are as follows

○ (A - λI)mx m = 0 & (A - λI)m-1x m ≠ 0

○ When m = 1, it is the same as the definition of eigenvector

○ Used in Jordan canonical form

② Example : The eigenvalue of the following matrix A is 1, and the corresponding eigenvector is v 1 = (1, 0)T

○ The algebraic multiplicity of eigenvalue 1 is 2, and the geometric multiplicity is 1

○ (A - λI)2 v 2 = (A - λI) {(A - λI) v 2} = 0 ⇔ (A - λI) v 2 = v 1

○ v 2 satisfies (A - I)v 2 **= (1, 0)T, and **v 2 = (*, 1)T (e.g., (0, 1)T) is called a generalized eigenvector

⑶ Eigenfunctions

① Theorem of Sturm-Liouville : Different eigenfunctions are orthogonal to each other

2. Diagonalization of Matrices

⑴ Diagonal Matrix

① An n × n matrix where all elements except the diagonal are zero

② Properties

○ Property 1. The determinant is the product of the diagonal elements

○ Property 2. The inverse matrix has the reciprocal of each diagonal element of the given diagonal matrix

⑵ Block Diagonal Matrix

① Definition : When Ai ∈ ℳni,ni (F) and n1 + ··· + nk = n, the matrix A in the following form

② Properties : When Ai, Bi ∈ ℳni,ni (F),

○ Property 1. diag(A1, ···, Ak)·diag(B1, ···, Bk) = diag(A1B1, ···, AkBk)

○ Property 2. If Ai is invertible, (diag(A1, ···, Ak))-1 = diag(A1-1, ···, Ak-1)

⑶ Diagonalizability

① Definition : A matrix A is diagonalizable if there exist a diagonal matrix D and an invertible matrix P satisfying the following

② Orthogonal Diagonalization : For a matrix P composed of orthonormal basis vectors

③ Theorem 1. The necessary and sufficient condition for an n × n matrix A to be diagonalizable is that A has n linearly independent eigenvectors

④ Theorem 2. If an n × n matrix A has n distinct eigenvalues λ1, λ2, ···, λn, it can be diagonalized

○ If P is a matrix having linearly independent eigenvectors of A as columns, D is as follows

⑤ Theorem 3. Spectral Theorem : An n × n matrix A can be orthogonally diagonalized if and only if A is a symmetric matrix

⑥ Theorem 4. Generalized eigenvectors of an n × n matrix A always span ℝn, thus such a matrix A can be diagonalized

○ Example : An n × n matrix satisfying A2 = A has eigenvalues λ = 0, 1 and can be diagonalized

⑷ Jordan Canonical Form

① When a matrix can’t be diagonalized, but there are still generalized eigenvectors corresponding to the eigenvalues, it can be transformed into the Jordan canonical form

② Jordan Canonical Form : A = PJP-1 (where J is not a complete diagonal matrix, but a matrix composed of Jordan blocks)

⑸ Examples

① Given matrix A

② Calculate eigenvalues and eigenvectors

③ Calculate P and D

3. Quadratic Forms

⑴ Definition : 4x12 + 2x1x2 + 3x22 is a quadratic form, but 4x12 + 2x1 is not a quadratic form

⑵ Representation using a symmetric matrix

⑶ Definition of signs of quadratic forms : For a symmetric matrix A ∈ ℳn and any vector x ≠ 0 on ℝn

① Positive definite matrix : All eigenvalues of A are greater than 0 ⇔ Q(x) > 0 for all x ≠ 0 ⇔

② Positive semi-definite matrix : All eigenvalues of A are greater than or equal to 0 ⇔ Q(x) ≥ 0 for all x ≠ 0 ⇔

③ Negative definite matrix : All eigenvalues of A are less than 0 ⇔ Q(x) < 0 for all x ≠ 0 ⇔

④ Negative semi-definite matrix : All eigenvalues of A are less than or equal to 0 ⇔ Q(x) ≤ 0 for all x ≠ 0 ⇔

⑤ Indefinite matrix : When the signs of the eigenvalues are not uniform, unlike ① to ④

⑥ Proof

○ Let A = PΛPT (where Λ is a diagonal matrix) and y = PTx, then,

○ x TAx = y TΛy, thus it is easy to confirm the relationship between the signs of the eigenvalues and x TAx

⑦ Application

⑷ Determination of the signs of quadratic forms : For a symmetric matrix A ∈ ℳn and a subset S =｛i1, i2, ···, ik｝ of ｛1, 2, ···, n｝

① Principal submatrix : A k × k matrix formed by selecting the rows and columns of A corresponding to S

② Leading principal submatrix : A k × k matrix obtained by omitting the last n-k row and column vectors of A

③ Theorem 1. A symmetric matrix A ∈ ℳn being positive definite is equivalent to sgn(det(Ak)) = 1, k = 1, 2, ···, n

④ Theorem 2. A symmetric matrix A ∈ ℳn being negative definite is equivalent to sgn(det(Ak)) = (-1)k, k = 1, 2, ···, n

⑤ Theorem 3. A symmetric matrix A ∈ ℳn being positive semi-definite is equivalent to all principal submatrix determinants being non-negative

○ (Note) If any leading principal submatrix determinant is 0 while attempting Theorem 1 or 2, try Theorem 3

○ Obviously, if the sign of any leading principal submatrix determinant changes, there is no need to attempt Theorem 3 or 4 as it indicates an indefinite matrix

⑥ Theorem 4. A symmetric matrix A ∈ ℳn being negative semi-definite is equivalent to all principal submatrix determinants of -A being non-negative

○ (Note) If the sign of any principal submatrix determinant is not uniform while attempting Theorem 3, try Theorem 4

⑦ (Reference) sgn indicates 1 for positive values and -1 for negative values when the determinant is directly calculated

⑸ For a C2 function f : Ω → ℝ defined on a region Ω ⊂ ℝn, if at point p ∈ Ω, ∇f(p) = 0, then the following holds

① If Hf(p) is a positive definite matrix, then f has a local minimum at point x = p

② If Hf(p) is a negative definite matrix, then f has a local maximum at point x = p

③ If Hf(p) is an indefinite matrix, then point x = p is a saddle point of f

④ If Hf(p) is a positive semi-definite matrix, then f has a relative minimum at point x = p

⑤ If Hf(p) is a negative semi-definite matrix, then f has a relative maximum at point x = p

① Theorem 1. If sgn(

Hm+i

) = (-1)m, i = m+1, m+2, ···, n, then QA(x) ＞ 0 holds

○ (Note) Keep in mind it is not a necessary and sufficient condition

② Theorem 2. If sgn(

Hm+i

) = (-1)i, i = m+1, m+2, ···, n, then QA(x) ＜ 0 holds

○ (Note) Keep in mind it is not a necessary and sufficient condition

③ Example : The following example observes a negative definite symmetric matrix A

④ (Reference) Even a quadratic form without a sign may have a sign under certain constraints

4. Differential Equations and Eigenvalues

⑴ Systems of Linear Ordinary Differential Equations : For systems of linear ordinary differential equations represented by a 2 × 2 matrix A,

① Case 1. If A has eigenvalues λ1, λ2 and is diagonalizable : The solution can be expressed as follows with eigenvectors v1, v2 corresponding to λ1, λ2

② Case 2. If A has complex eigenvalues p ± qi : The solution can be expressed as follows

③ Case 3. If A has a single eigenvalue λ with multiplicity 2 and is not diagonalizable, the general solution x together with generalized eigenvectors is as follows

⑵ Linear Approximation

① Example : For x’ = x3 - y, y’ = x, the classification of the equilibrium point (0, 0)

② Step 1. The following approximations can be found

③ Step 2. Construct the Jacobian matrix

④ Step 3. Calculate eigenvalues

○ When all eigenvalues are negative real numbers : The equilibrium point is stable

○ When one eigenvalue is positive and the other is negative : The equilibrium point is a saddle

○ When all eigenvalues are positive real numbers : The equilibrium point is unstable

○ When all eigenvalues are complex numbers : The equilibrium point is a spiral

○ Since A’s eigenvalues are ± i, a spiral pattern appears around (0, 0)

Figure 1. Stream plot of the given system

Input: 2020.05.19 23:48

Modified: 2024.11.12 09:45