Eigenvalues, spectral geometry, and quadratic forms

A repeated linear transformation can stretch one direction, reverse another, rotate a plane or mix directions through a shear. Eigenvectors identify directions that the map leaves invariant up to scalar multiplication. When enough independent eigenvectors exist, their coordinates turn the map into separate scalar actions. A real symmetric matrix has an especially strong version: an orthonormal eigenbasis, real eigenvalues and a direct interpretation of quadratic geometry.

Those conclusions do not hold for every matrix. A quarter-turn has no real eigenvector. A defective two-dimensional map has too few independent eigenvectors even though its characteristic equation has two roots counting multiplicity. A nonsymmetric map can amplify a norm temporarily while every eigenvalue has magnitude below one. This lesson teaches the useful spectral results together with the assumptions that make each conclusion valid.

Retrieval check: explain orthonormal coordinates and a similarity transformation P⁻¹AP. Review Module 12 and basis changes. Determinants and inverse maps come from Module 10. Complex numbers are introduced below where needed; no calculus is required for the finite-dimensional arguments.

For a square linear map A, an eigenvector v is a nonzero vector satisfying Av=λv for some scalar eigenvalue λ. Together they form an eigenpair. Nonzero is essential: zero satisfies the equation for every λ and therefore identifies no special direction. Every nonzero scalar multiple of an eigenvector is another eigenvector for the same value. The invariant object is usually its line or larger eigenspace, not one uniquely signed and normalised coordinate array.

If λ>0 in a real eigenpair, A preserves the ray’s orientation and scales its length by λ. Negative λ reverses orientation, and λ=0 collapses that eigenvector to zero. An eigenvalue equal to one fixes that entire eigendirection. The eigenvalue need not describe the scaling of every vector, since other vectors can combine several eigendirections or the map can lack a complete eigenbasis. Eigenvectors must also belong to the same input/output space; a rectangular map does not have this ordinary equation with arbitrary equal-shaped v unless another construction is specified.

Rearranging gives (A−λI)v=0. A nonzero solution exists exactly when the square matrix A−λI is singular, equivalently det(A−λI)=0. This is the characteristic equation. For A=[[a,b],[c,d]], it is (a−λ)(d−λ)−bc=λ²−(a+d)λ+(ad−bc)=0. Thus the two roots, counting multiplicity over the complex numbers, have sum trace(A)=a+d and product det(A)=ad−bc. These checks help detect arithmetic mistakes but do not replace verifying each corresponding vector.

The eigenspace E_λ is the kernel N(A−λI), including zero as a subspace even though zero is not itself an eigenvector. A basis for that kernel gives all eigenvectors for λ through nonzero combinations. Its dimension is the geometric multiplicity. The number of times λ appears as a characteristic-polynomial root is the algebraic multiplicity. They are different quantities; repeated roots do not automatically supply additional independent directions. A scalar matrix λI has a repeated root and every vector as an eigenvector, while the defective example below has a repeated root and only one line.

Eigenvalues are invariant under a basis change. If B=P⁻¹AP and Av=λv, then B(P⁻¹v)=P⁻¹Av=λ(P⁻¹v). The transformed vector is nonzero because P is invertible, and reversing the change gives the converse. The coordinate tuples can differ, but the map’s spectral values do not. Some simple modifications can also be predicted directly: A+αI has the same eigenvectors with values λ+α, and an invertible A has inverse eigenvalues 1/λ. Verify the latter by applying A⁻¹ to Av=λv, noting that invertibility excludes λ=0. These identities are useful checks when analysing a shifted system or an inverse solve.

To compute a small eigenpair, first solve the characteristic equation, then solve each homogeneous system, and finally substitute into Av=λv. Choosing a unit length makes comparisons convenient but does not remove sign ambiguity. In floating-point calculations, inspect a scaled residual such as ‖Av−λv‖ relative to ‖A‖‖v‖ and the intended tolerance. A numerically tiny residual validates the pair for the supplied array; it does not establish that eigenvectors are unique or that a nearby repeated spectrum has been resolved reliably.

If A has d independent eigenvectors in a d-dimensional space, put them into an invertible matrix V and their eigenvalues into diagonal Λ in the same order. The column equations give AV=VΛ, hence A=VΛV⁻¹. This is diagonalisation. In eigenbasis coordinates c=V⁻¹x, one application becomes Λc. The columns need not be orthogonal; replacing V⁻¹ by Vᵀ is valid only when V is orthogonal. Conversely any such diagonal representation supplies an eigenbasis from its columns, so the existence of a full independent eigenbasis is exactly the condition.

Eigenvectors for distinct eigenvalues are independent. A short proof uses a minimal dependent sublist. Suppose Σcᵢvᵢ=0 with every coefficient in that minimal sublist nonzero. Apply A then subtract λ_last times the original relation. The last term vanishes and each remaining coefficient becomes cᵢ(λᵢ−λ_last), which is nonzero because the values are distinct. This is a shorter dependence relation, a contradiction. Thus d distinct eigenvalues guarantee diagonalisation over the field containing them. Repeated values require checking the dimensions of their eigenspaces instead of assuming the same conclusion.

Complex numbers allow characteristic roots that are not real. Write i²=−1 and z=a+bi; addition and multiplication follow this rule, and |z|=√(a²+b²). Real matrices can act on complex vectors by the same entry formulas. A quarter-turn R=[[0,−1],[1,0]] has characteristic equation λ²+1=0 and values i,−i. For i, a vector (1,−i) satisfies R(1,−i)=(i,1)=i(1,−i). No nonzero real v satisfies Rv=λv with a real λ, since rotating by ninety degrees does not keep a real line invariant.

The quarter-turn is diagonalizable over the complex field, but it has no real eigenbasis. This distinction should be stated when saying “diagonalizable.” Complex conjugate roots come in pairs for real characteristic polynomials; conjugating an eigenpair also produces an eigenpair. A real two-dimensional rotation can therefore be described by a real rotation block or by two complex scalar modes. The complex magnitudes one explain preservation of amplitude; they do not mean each real coordinate remains fixed.

Repeated eigenvalues can also introduce basis ambiguity even when diagonalisation succeeds. For A=I₂ every nonzero vector is an eigenvector with value one. Any independent basis diagonalises it, and any orthonormal basis is equally valid for its symmetric spectral representation. A library can rotate the basis inside this repeated eigenspace. Comparing its individual columns with one preferred answer is consequently the wrong correctness test; compare the eigenspace or reconstructed map instead.

Diagonalisation is a change of description. It can simplify a map’s powers and geometry, but it does not automatically make the coordinate change numerically safe. A nearly dependent eigenvector matrix V can strongly distort lengths in its coordinates, and an exact defective map has no invertible V at all. Numerical output that presents almost parallel columns should not be accepted as a robust independent basis merely because the array has two columns.

The real symmetric spectral theorem states that A=Aᵀ has an orthonormal basis of real eigenvectors, so A=QΛQᵀ with real diagonal Λ and orthogonal Q. This is stronger than arbitrary diagonalisation: Q⁻¹=Qᵀ preserves Euclidean lengths, and the eigenvalues can be ordered as real numbers. Symmetry is a hypothesis to verify. A routine that reads only one triangle of an array can return a valid decomposition of the symmetric matrix implied by that triangle without solving the nonsymmetric array you intended.

Distinct-eigenvalue eigenvectors of a symmetric matrix are perpendicular. If Au=λu and Av=μv, then λuᵀv=(Au)ᵀv=uᵀAv=μuᵀv. Thus (λ−μ)uᵀv=0, and λ≠μ forces uᵀv=0. Inside a repeated eigenspace, any linear combination remains an eigenvector, so Gram–Schmidt can create an orthonormal basis without changing its eigenvalue. These arguments explain orthogonality once eigenvectors exist; existence of enough real directions needs an additional argument.

Here is a finite-dimensional proof, with its analysis input stated explicitly. A continuous real function on the compact unit sphere attains a maximum; this standard compactness result is used here, rather than proved before the limits and continuity block. Apply it to q(x)=xᵀAx to obtain a unit maximiser v with value λ. For any u perpendicular to v and any real t, the normalised candidate (v+tu)/√(1+t²‖u‖²) has quadratic value no larger than λ. Expanding and cancelling yields 2t uᵀAv+t²(uᵀAu−λ‖u‖²)≤0 for every t. If uᵀAv were nonzero, a sufficiently small t of the same sign would make the linear term dominate the quadratic term and violate this inequality. Hence uᵀAv=0 for every u perpendicular to v.

It follows that Av is parallel to v, and dotting with v gives Av=λv. The perpendicular complement of v is invariant: for w⊥v, vᵀAw=(Av)ᵀw=λvᵀw=0. The restriction of A to this (d−1)-dimensional complement is still symmetric in any orthonormal basis of it. By induction on dimension, starting from a one-dimensional real scalar map, the restriction has an orthonormal real eigenbasis. Joining v gives one for the entire space. This proves the spectral theorem from the declared compactness fact and previously established orthonormal-basis construction, with no differentiation.

In general A=Σλᵢqᵢqᵢᵀ. Each qᵢqᵢᵀ projects onto an eigendirection and λᵢ scales it. For a repeated value, sum the projectors over its whole eigenspace; that sum is basis independent. Individual signed vectors are not. This is useful for validating decompositions and interpreting geometry without attaching significance to arbitrary column signs or rotations within a repeated subspace.

Orthogonal diagonalisation also characterises real symmetry in the reverse direction: if A=QΛQᵀ with real diagonal Λ, transposing gives Aᵀ=QΛQᵀ=A. Thus a nonsymmetric real matrix cannot have this representation, even if all its eigenvalues happen to be real. The theorem is a structural result about self-adjoint Euclidean maps, not a general promise made by an eigenvalue solver.

A quadratic form is q(x)=xᵀAx. For a real nonsymmetric A, only its symmetric part S=(A+Aᵀ)/2 contributes: the skew part K=(A−Aᵀ)/2 satisfies xᵀKx=0 because this scalar equals its own transpose xᵀKᵀx=−xᵀKx. Thus quadratic classification concerns S, not arbitrary nonsymmetric eigenvalues of A. In the main discussion A is symmetric, making the spectral theorem directly applicable.

Write x=Qc and use QᵀQ=I. Then xᵀAx=cᵀΛc=Σλᵢcᵢ². A is positive definite if this value is strictly positive for every nonzero x, exactly when every λᵢ>0. It is positive semidefinite if it is nonnegative for every x, exactly when every λᵢ≥0. Negative definite and semidefinite reverse the signs. It is indefinite when both positive and negative values occur, exactly when there are eigenvalues of both signs. Zero eigenvalues create flat directions and prevent strict definiteness.

The equivalences follow in both directions: the sign conditions make every squared-coordinate sum have the claimed sign, while an eigenvector with a forbidden-sign eigenvalue supplies a counterexample. For a positive-semidefinite matrix, q(x)=0 exactly when x has components only in its zero eigenspaces, equivalently Ax=0. This implication fails for indefinite forms: positive and negative terms can cancel at a nonzero x while Ax remains nonzero. A zero quadratic value must therefore be interpreted with the definiteness assumption.

For a positive-definite form in d dimensions, the q=1 surface has principal semiaxis lengths 1/√λᵢ, not λᵢ. Large eigenvalues penalise movement strongly and make the corresponding contour axis short. The transformation A itself scales eigenvector lengths by λᵢ, which is a different geometric statement. Confusing the map’s stretching factors with its quadratic level-set radii reverses the contour interpretation.

These forms appear in squared errors, energy functions and later curvature matrices. A Gram matrix XᵀX is always positive semidefinite because xᵀXᵀXx=‖Xx‖²≥0. It is positive definite exactly when X has full column rank. That conclusion can be proved directly before computing eigenvalues; the spectrum then supplies direction strengths. A symmetric matrix with positive entries is not necessarily positive definite: [[1,2],[2,1]] has eigenvalues three and minus one. Entrywise positivity and quadratic positivity are different properties.

An optional connection to undirected graphs is the Laplacian L=D−G, where G is a symmetric adjacency matrix with nonnegative edge weights and D’s diagonal contains their row sums. Expanding gives xᵀLx=Σ_{i<j}Gᵢⱼ(xᵢ−xⱼ)², so L is positive semidefinite. The constant vector has eigenvalue zero because each row sum of L is zero. Zero energy requires equal values across every positive-weight edge, hence values constant on each connected component. Conversely such vectors have zero energy and belong to the kernel. Therefore the zero eigenspace dimension counts components. This spectral fact has a concrete proof from an energy expression; it does not require guessing eigenvalues from a drawing.

For numerical classification, specify how close-to-zero values are treated, and do not treat rounded negative values as automatically meaningful physical instability. The exact symmetric matrix, its measurement uncertainty and arithmetic error may give different levels of certainty about the sign of a tiny eigenvalue. The explorer uses deliberately separated preset spectra so its exact classifications are clear; later numerical computation develops the more delicate cases.

For e_{k+1}=Be_k, induction gives e_k=B^k e₀. In a diagonalizable case B=VΛV⁻¹, cancelling adjacent inverse factors gives B^k=VΛ^kV⁻¹. Each eigenmode has coefficient λᵢ^k times its initial coefficient. Magnitude below one decays geometrically; above one grows when that mode is excited; magnitude one need not decay and may oscillate. A negative real value alternates sign, while a complex value combines amplitude scaling and rotation.

The spectral radius ρ(B) is the largest eigenvalue magnitude, including complex values. For a diagonalizable finite matrix, ρ<1 makes every initial state tend to zero. If some value has magnitude at least one, its eigenvector provides a trajectory that does not tend to zero. For complex vectors of a real matrix, at least one of the real and imaginary parts must fail to tend to zero if the complex trajectory fails, so this obstruction also concerns real initial states. “Every initial state” distinguishes global asymptotic decay from a single trajectory that does not excite an unstable mode.

For symmetric B, orthonormal coordinates give the stronger estimate ‖B^k e₀‖≤ρ(B)^k‖e₀‖. If ρ<1, Euclidean norm contracts at every step. For general diagonalizable B, a bound also includes the distortion factors of V and V⁻¹, so decay need not be monotone in ordinary coordinates. Nonsymmetric mixing can amplify some vectors before eventual decay. An estimate of an asymptotic rate is not the same as a one-step Euclidean contraction guarantee.

Defective maps require more than the simple eigenbasis power formula. Let N=[[0,3],[0,0]], so N²=0, and B=0.5I+N. The binomial expansion terminates: for k≥1, B^k=0.5^k I+k0.5^{k−1}N. At e₀=(0,1), the first step is (3,0.5), larger in norm than the initial unit vector, but both coordinates eventually decay. The factor k delays geometric decay without preventing it. At B=I+[[0,1],[0,0]], the analogous formula gives B^k e₀=(k,1), unbounded although the spectral radius is one. Equality at one is a boundary requiring further analysis.

The full finite-dimensional theorem states B^k→0 exactly when ρ(B)<1, even for defective matrices. Its general sufficiency proof uses the Jordan-form existence theorem, whose canonical-form construction is beyond this lesson. Within a Jordan block, powers are finite sums of binomial coefficients times λ^{k−j}N^j, where N is nilpotent. Each term is a polynomial in k times a decaying geometric factor when |λ|<1; zero-eigenvalue blocks vanish after finitely many powers. Such products tend to zero, as seen from their eventual ratio approaching |λ|<1. Combining finitely many blocks and a fixed coordinate change preserves convergence. Necessity follows from the eigenvector obstruction above. This explains the theorem’s mechanism and explicitly identifies the additional canonical-form result used.

The weaker condition ρ≤1 is not a universal boundedness guarantee, as the unit defective example demonstrates. Conversely a quarter-turn has ρ=1 and bounded norms but does not converge to zero. Boundedness, monotone contraction and asymptotic decay are different behaviours. In an affine iteration x_{k+1}=Bx_k+c, subtract a fixed point x* satisfying x*=Bx*+c to obtain this error equation. If no fixed point has been established, one cannot simply identify x_k itself with the homogeneous error. These distinctions prepare for optimisation and iterative algorithms later in the course.

For nonzero x and real symmetric A, the Rayleigh quotient is R_A(x)=(xᵀAx)/(xᵀx). With x=Qc, it is Σλᵢcᵢ²/Σcᵢ², a weighted average of the eigenvalues with nonnegative weights summing to one. Therefore λ_min≤R_A(x)≤λ_max. Equality at an extreme occurs when x lies entirely in its extreme eigenspace. This proves the bounds and their attainment without differentiating a constrained objective.

The largest quotient identifies a direction of greatest quadratic energy per unit squared length. The smallest identifies the least. For a repeated extreme value, there is an entire subspace of maximising or minimising directions, not one preferred vector. For a nonsymmetric real matrix, the quotient describes its symmetric part, so it should not be used to claim those same ordered bounds for possibly complex eigenvalues of the original map. Match the quotient’s structural hypothesis to the intended interpretation.

The Euclidean operator norm is ‖A‖₂=max_{‖x‖=1}‖Ax‖. For symmetric A, squared length in eigen-coordinates is Σλᵢ²cᵢ², so the norm is max|λᵢ|, attained on a corresponding eigenvector. For symmetric invertible A, its inverse has eigenvalues 1/λᵢ and norm 1/min|λᵢ|. Thus κ₂(A)=‖A‖₂‖A⁻¹‖₂=max|λᵢ|/min|λᵢ|. In the positive-definite case every value is positive, giving λ_max/λ_min. A singular matrix has no finite ordinary inverse-condition ratio.

For Ax=b with nonzero b, a RHS change δb changes the exact solution by δx=A⁻¹δb. The norm bounds give ‖δx‖≤‖A⁻¹‖‖δb‖ and ‖b‖=‖Ax‖≤‖A‖‖x‖. Combining them yields ‖δx‖/‖x‖≤κ₂(A)‖δb‖/‖b‖. This is an upper bound, not a statement that every perturbation attains the amplification. It concerns RHS changes for a fixed invertible A; matrix perturbations require an additional analysis. The direction of the disturbance matters as well as its size.

For a nonsymmetric matrix, eigenvalue magnitudes do not generally give the Euclidean condition number. The unit defective shear [[1,1],[0,1]] has both eigenvalues one but distorts lengths and has condition number larger than one. General condition analysis uses singular values, which Module 14 develops. Similarly, a large or small spectral radius describes repeated-map modes, not automatically conditioning of a one-time solve or a statistical uncertainty estimate.

In later lessons, covariance and symmetric curvature matrices will admit this principal-direction interpretation when their assumptions are met. Here the algebra establishes which directions have high energy, which are flat and which are sensitive in an inverse calculation. It does not say that the largest-variance direction is the best classifier, or that an eigenvector is a causal feature. Numerical comparisons should allow sign changes, eigenvalue ordering and rotations inside repeated eigenspaces, and should verify eigenpair residuals, orthogonality for symmetric inputs, and reconstruction. Interpretation begins after those structural checks, not merely after a solver returns.

A simple power iteration repeatedly applies A and normalises to approximate a dominant eigendirection. For a symmetric A with one eigenvalue of strictly largest magnitude and a starting vector containing that mode, divide its spectral expansion after k steps by the dominant factor. Every other coefficient is multiplied by (λᵢ/λ_dom)^k and decays. This proves direction convergence up to sign. A negative dominant value can make the signed unit vector alternate, and equal largest magnitudes can prevent selection of a unique direction. Starting perpendicular to the dominant eigenspace also defeats that claim. Thus an attractive short numerical algorithm still needs a spectral gap and a nonzero starting component; its stopping criterion should inspect an eigenpair residual rather than just one pair of nearly equal signed arrays.

Use the Python 3.11 and NumPy environment from array preparation. Outputs are captured by executing each downloadable script. Validate equations and subspaces with tolerances; the exact examples supply the mathematical classification independently of rounded output.

Lab A Symmetric eigenpairs and reconstruction

Predict the eigenvalues and unit eigenvectors before running. Compare columns allowing sign changes, verify A Q=QΛ,QᵀQ=I and QΛQᵀ=A, then evaluate two Rayleigh quotients. Explain why a rotated basis is also valid for the repeated spectrum of the identity.

Download lab1_symmetric_eigenpairs.py

"""Check eigenpairs and subspaces rather than demanding one vector sign."""
import numpy as np
np.set_printoptions(precision=6, suppress=True)
A = np.array([[2., 1.], [1., 2.]])
values, vectors = np.linalg.eigh(A)
hand = np.array([[1., 1.], [-1., 1.]])/np.sqrt(2)
assert np.allclose(values, [1., 3.])
assert np.allclose(A@vectors, vectors*values)
assert np.allclose(vectors.T@vectors, np.eye(2))
assert np.allclose(vectors@np.diag(values)@vectors.T, A)
print("Eigenvalues:", values)
print("Unit eigenvectors in columns:", vectors.tolist())
print("Absolute alignment with hand basis:", np.abs(hand.T@vectors).tolist())
for label, vector in [("difference", np.array([1., -1.])), ("sum", np.array([1., 1.]))]:
    quotient = vector@A@vector/(vector@vector)
    print("Rayleigh quotient:", label, quotient)
    assert quotient == (1 if label == "difference" else 3)
angle = np.pi/5
rotation = np.array([[np.cos(angle), -np.sin(angle)], [np.sin(angle), np.cos(angle)]])
identity = np.eye(2)
assert np.allclose(identity@rotation, rotation)
assert np.allclose(rotation.T@rotation, np.eye(2))
print("Repeated eigenvalue identity admits the rotated orthonormal basis:", True)
print("Positive-definite condition ratio:", values[-1]/values[0])

Eigenvalues: [1. 3.]
Unit eigenvectors in columns: [[-0.7071067811865475, 0.7071067811865475], [0.7071067811865475, 0.7071067811865475]]
Absolute alignment with hand basis: [[0.9999999999999998, 0.0], [0.0, 0.9999999999999998]]
Rayleigh quotient: difference 1.0
Rayleigh quotient: sum 3.0
Repeated eigenvalue identity admits the rotated orthonormal basis: True
Positive-definite condition ratio: 3.0

Deliverable: hand eigenpairs, numerical residual/reconstruction checks, the quadratic classification and the SPD condition ratio. Distinguish a sign change from a different eigenspace.

Lab B Linear iterations and scope of stability

Predict each behaviour from the matrix structure. Compare an excited unstable direction with an unexcited one. For the defective stable case, derive B^k=0.5^kI+k0.5^{k−1}N, then compare initial amplification and eventual decay. Explain what the unit defective and rotation cases show about the boundary ρ=1.

Download lab2_linear_iterations.py

"""Asymptotic decay does not require monotone Euclidean norm decay."""
import numpy as np
cases = [
    ("symmetric stable", np.diag([0.5, -0.8]), np.array([1., 1.])),
    ("unstable excited", np.diag([1.2, 0.5]), np.array([1., 1.])),
    ("unstable but unexcited mode", np.diag([1.2, 0.5]), np.array([0., 1.])),
    ("defective stable transient", np.array([[0.5, 3.], [0., 0.5]]), np.array([0., 1.])),
    ("unit defective growth", np.array([[1., 1.], [0., 1.]]), np.array([0., 1.])),
    ("rotation unit modulus", np.array([[0., -1.], [1., 0.]]), np.array([1., 0.])),
]
for label, B, initial in cases:
    radius = max(abs(np.linalg.eigvals(B)))
    print(label, "spectral radius:", float(radius))
    for step in [0, 1, 2, 5, 10, 20]:
        value = np.linalg.matrix_power(B, step)@initial
        print(" k / vector / norm:", step, [round(float(v), 9) for v in value], format(np.linalg.norm(value), ".6e"))
    if label == "defective stable transient":
        assert np.linalg.norm(B@initial) > np.linalg.norm(initial)
        assert np.linalg.norm(np.linalg.matrix_power(B, 20)@initial) < 0.001
    if label == "unit defective growth":
        assert np.allclose(np.linalg.matrix_power(B, 20)@initial, [20, 1])
print("Global asymptotic decay concerns every initial state, not one unexcited trajectory.")

symmetric stable spectral radius: 0.8
 k / vector / norm: 0 [1.0, 1.0] 1.414214e+00
 k / vector / norm: 1 [0.5, -0.8] 9.433981e-01
 k / vector / norm: 2 [0.25, 0.64] 6.870953e-01
 k / vector / norm: 5 [0.03125, -0.32768] 3.291667e-01
 k / vector / norm: 10 [0.000976562, 0.107374182] 1.073786e-01
 k / vector / norm: 20 [9.54e-07, 0.011529215] 1.152922e-02
unstable excited spectral radius: 1.2
 k / vector / norm: 0 [1.0, 1.0] 1.414214e+00
 k / vector / norm: 1 [1.2, 0.5] 1.300000e+00
 k / vector / norm: 2 [1.44, 0.25] 1.461540e+00
 k / vector / norm: 5 [2.48832, 0.03125] 2.488516e+00
 k / vector / norm: 10 [6.191736422, 0.000976562] 6.191736e+00
 k / vector / norm: 20 [38.337599924, 9.54e-07] 3.833760e+01
unstable but unexcited mode spectral radius: 1.2
 k / vector / norm: 0 [0.0, 1.0] 1.000000e+00
 k / vector / norm: 1 [0.0, 0.5] 5.000000e-01
 k / vector / norm: 2 [0.0, 0.25] 2.500000e-01
 k / vector / norm: 5 [0.0, 0.03125] 3.125000e-02
 k / vector / norm: 10 [0.0, 0.000976562] 9.765625e-04
 k / vector / norm: 20 [0.0, 9.54e-07] 9.536743e-07
defective stable transient spectral radius: 0.5
 k / vector / norm: 0 [0.0, 1.0] 1.000000e+00
 k / vector / norm: 1 [3.0, 0.5] 3.041381e+00
 k / vector / norm: 2 [3.0, 0.25] 3.010399e+00
 k / vector / norm: 5 [0.9375, 0.03125] 9.380207e-01
 k / vector / norm: 10 [0.05859375, 0.000976562] 5.860189e-02
 k / vector / norm: 20 [0.000114441, 9.54e-07] 1.144449e-04
unit defective growth spectral radius: 1.0
 k / vector / norm: 0 [0.0, 1.0] 1.000000e+00
 k / vector / norm: 1 [1.0, 1.0] 1.414214e+00
 k / vector / norm: 2 [2.0, 1.0] 2.236068e+00
 k / vector / norm: 5 [5.0, 1.0] 5.099020e+00
 k / vector / norm: 10 [10.0, 1.0] 1.004988e+01
 k / vector / norm: 20 [20.0, 1.0] 2.002498e+01
rotation unit modulus spectral radius: 1.0
 k / vector / norm: 0 [1.0, 0.0] 1.000000e+00
 k / vector / norm: 1 [0.0, 1.0] 1.000000e+00
 k / vector / norm: 2 [-1.0, 0.0] 1.000000e+00
 k / vector / norm: 5 [0.0, 1.0] 1.000000e+00
 k / vector / norm: 10 [-1.0, 0.0] 1.000000e+00
 k / vector / norm: 20 [1.0, 0.0] 1.000000e+00
Global asymptotic decay concerns every initial state, not one unexcited trajectory.

Deliverable: distinguish global asymptotic decay, one-trajectory decay, boundedness and monotone norm contraction. A finite table illustrates these behaviours; the formulas prove the infinite-time statements.

Lab C Signs defects and solver assumptions

Flip a symmetric eigenvector’s sign and compare its projector. Diagnose J’s exact one-dimensional eigenspace, then inspect the near-parallel columns returned numerically. Deliberately pass a nonsymmetric J to the symmetric solver and verify its output against the original J. Finally check the valid complex eigenpairs of a real quarter-turn.

Download lab3_eigenvector_faults.py

"""Signs and repeated eigenspaces are ambiguous; symmetry is a real contract."""
import numpy as np
A = np.array([[2., 1.], [1., 2.]])
values, vectors = np.linalg.eigh(A)
flipped = vectors.copy()
flipped[:, 0] *= -1
print("Direct vector equality after sign flip:", np.allclose(vectors, flipped))
print("First eigenspace projector equality:", np.allclose(np.outer(vectors[:, 0], vectors[:, 0]), np.outer(flipped[:, 0], flipped[:, 0])))
assert np.allclose(A@flipped, flipped*values)

J = np.array([[1., 1.], [0., 1.]])
j_values, j_vectors = np.linalg.eig(J)
print("Defective eigenvalues:", j_values.tolist())
print("Reported vector-matrix rank at tolerance 1e-12:", np.linalg.matrix_rank(j_vectors, tol=1e-12))
print("Exact eigenspace constraint from (J-I)v=0: v2=0; dimension one")
assert np.linalg.matrix_rank(j_vectors, tol=1e-12) == 1
assert np.array_equal(J-np.eye(2), [[0., 1.], [0., 0.]])
wrong_values, wrong_vectors = np.linalg.eigh(J)
wrong_residual = np.linalg.norm(J@wrong_vectors-wrong_vectors*wrong_values)
print("Misused eigh eigenpair residual on original J:", wrong_residual)
assert wrong_residual > 0.5
print("Symmetry guard:", np.array_equal(J, J.T))
assert not np.array_equal(J, J.T)

rotation = np.array([[0., -1.], [1., 0.]])
r_values, r_vectors = np.linalg.eig(rotation)
print("Rotation eigenvalues:", [str(v) for v in r_values])
assert np.allclose(rotation@r_vectors, r_vectors*r_values)
assert np.allclose(np.abs(r_values), 1)
print("No real eigenbasis for a quarter-turn; complex eigenpairs are valid, not a solver failure.")

Direct vector equality after sign flip: False
First eigenspace projector equality: True
Defective eigenvalues: [1.0, 1.0]
Reported vector-matrix rank at tolerance 1e-12: 1
Exact eigenspace constraint from (J-I)v=0: v2=0; dimension one
Misused eigh eigenpair residual on original J: 1.0
Symmetry guard: False
Rotation eigenvalues: ['1j', '-1j']
No real eigenbasis for a quarter-turn; complex eigenpairs are valid, not a solver failure.

Deliverable: repair a direct-array comparison, reject a purported complete eigenbasis for J, and state the correct symmetry contract. Explain why output existing is not sufficient evidence that the intended problem was solved.

Exercises 1–12 are required; 13–14 add 35 optional minutes. State the scalar field, symmetry and initial-state assumptions whenever they matter.

Use eigenvalues, symmetric decompositions and quadratic geometry in the authors’ Mathematics for Machine Learning companion. Numerical contracts are in the official NumPy 1.26 eig and eigh documentation. The original arguments above label the compactness input to the spectral proof and the canonical-form input to general defective stability.

Without notes, verify a two-dimensional eigenpair and prove the diagonal representation when a basis exists. State the symmetric theorem and its proof inputs. Classify a form, draw its principal axes, and explain each stability and condition-number hypothesis separately.

Exit task: A=[[3,1],[1,3]] has eigenvalues two on the difference direction and four on the sum direction. Its unit quadratic contour has semiaxes 1/√2 and 1/2; κ₂=2. For B=A/5,ρ=4/5 and every Euclidean error norm contracts by at most 4/5. Explain why each conclusion used symmetry and why replacing A by a general nonsymmetric matrix invalidates the same unqualified formulas.

Ready to move on: you can recognise which spectral conclusions are justified and verify the representation that supports them. Module 14 develops singular values for rectangular and nonsymmetric maps, low-rank approximation, the pseudoinverse and centred PCA. Check the course overview for availability.