Read a self-adjoint operator through real energy values

Try

For the symmetric matrix A=[[3,1],[1,3]], compare the unit directions q₁=(1,−1)/√2 and q₂=(1,1)/√2. Direct multiplication gives Aq₁=2q₁ and Aq₂=4q₂. The two perpendicular directions have different amplification factors. A general unit vector x=c₁q₁+c₂q₂ has quadratic value ⟨Ax,x⟩=2|c₁|²+4|c₂|², with |c₁|²+|c₂|²=1. Its value must lie between two and four.

Predict

Predict which vectors attain the endpoints. A weighted average reaches two only when the coefficient on the four-direction vanishes, and reaches four only when the coefficient on the two-direction vanishes. This is more precise than saying that eigenvectors are “important directions.” They are exact extremizers of a scalar measurement on the unit sphere. If an extreme eigenvalue repeats, every nonzero vector in its whole eigenspace attains that extreme.

Experiment

Make a table for x=q₁, q₂ and (q₁+q₂)/√2. Their Rayleigh quotients are 2,4 and 3. Verify the middle value by using the standard-coordinate vector (1,0), which is the third vector in this list. Then move along x=cos θ q₁+sin θ q₂ and calculate R(x)=2+2sin²θ. Sketch this function over a half-turn and mark the minimum and maximum. The curve explains why merely checking a few coordinate directions can miss the true extrema.

Understand

Self-adjoint means T*=T. If Tv=λv, then λ‖v‖²=⟨Tv,v⟩=⟨v,Tv⟩=conj(λ)‖v‖², so λ is real. A complex self-adjoint operator is normal and hence has an orthonormal eigenbasis by the normal spectral theorem. For a real symmetric operator, complexify the matrix to obtain real eigenvalues; a complex eigenvector has real and imaginary parts satisfying the same real eigenvalue equation, at least one nonzero. Choose a real unit eigenvector. Its real orthogonal complement is invariant because ⟨Tx,v⟩=⟨x,Tv⟩=0 there. Induction produces a real orthonormal eigenbasis. This proves orthogonal diagonalization over R rather than merely importing a complex basis.

Apply

For nonzero x define the Rayleigh quotient R(x)=⟨Tx,x⟩/‖x‖². Expand in an orthonormal eigenbasis to obtain a weighted average of real eigenvalues. This proves its bounds and equality conditions. Conversely, the quotient can characterize extreme eigenvalues variationally. As a proof exercise, minimize it on the unit sphere, whose compactness guarantees an extremizer, and examine variations in every perpendicular direction; the resulting orthogonality forces Tx to be a multiple of x. The spectral argument already supplies the result, while this alternate route connects linear algebra to constrained optimization.

One variational consequence concerns restriction to a subspace. If U is invariant for self-adjoint T, then U⊥ is invariant too: for x∈U⊥ and u∈U, the inner product ⟨Tx,u⟩ equals ⟨x,Tu⟩ and vanishes. The matrix is therefore block diagonal in adapted orthonormal bases. Contrast this with the nilpotent counterexample from the invariant-subspace lesson, where no invariant complement existed. Self-adjointness supplies the additional structure that repairs that failure.

Reconstruct from prescribed images, without assuming consistency

Suppose measurements prescribe Txᵢ=yᵢ on finitely many vectors in a finite-dimensional real or complex inner-product space V. Before asking for self-adjointness, check every linear relation among the inputs: whenever Σcᵢxᵢ=0, the outputs must satisfy Σcᵢyᵢ=0. This condition is necessary, and it is sufficient to define a linear map T₀ on U=span(xᵢ) by T₀(Σcᵢxᵢ)=Σcᵢyᵢ. Two representations differ by a relation, so the value is independent of the representation. In practice choose a basis from the input list and check each remaining input against its expansion.

The second condition is ⟨yᵢ,xⱼ⟩=⟨xᵢ,yⱼ⟩ for every pair. Sesquilinearity extends these equalities to ⟨T₀u,v⟩=⟨u,T₀v⟩ for all u,v in U. This does not require T₀U to lie in U: only its compression back into U is forced to be self-adjoint. Do not silently replace compression symmetry by invariance.

Check the data before completing the matrix

The inputs are x₁=e₁, x₂=e₁+e₂ and x₃=x₁+x₂. Their outputs must respect that relation. Self-adjointness imposes a separate pairwise inner-product test. Predict which change prevents every extension, and which changes only an unobserved direction.

First coordinate of y₂ 234 Fault in the third coordinate of y₃ 0: follows the input relation1: breaks the input relation Free complement coefficient γ -202

Compatible: the full family has one free real parameter.

Index	Input	Prescribed output	Symmetry residuals against all inputs
1	(1, 0, 0)	(1, 2, 1)	0, 0, 0
2	(1, 1, 0)	(3, 4, -1)	0, 0, 0
3	(2, 1, 0)	(4, 6, 0)	0, 0, 0

Completed matrix rows: (1, 2, 1); (2, 2, -2); (1, -2, 0).

The unobserved e₃ maps to (1, -2, 0). Changing γ preserves every prescribed output. The observed plane is not invariant: its images can have a third component. Thus its perpendicular axis need not be an eigenvector.

Reset

This exact integer experiment has 18 bounded control combinations. The unrestricted family permits every real γ; the selector samples three values. State is temporary and awards no mastery.

Prove sufficiency and describe every extension

Use the orthogonal decomposition V=U⊕U⊥ and let P be projection onto U. Write T₀u=Au+Bu, where A=PT₀ maps U to U and B=(I−P)T₀ maps U to U⊥. The pairwise condition says exactly that A is self-adjoint. In adapted orthonormal coordinates any extension must have its first block column fixed as (A,B). Self-adjointness then forces the upper-right block to be B*, while the remaining block C on U⊥ may be any self-adjoint map. Conversely the block map with rows (A,B*) and (B,C) is self-adjoint and agrees with all prescribed data. This proves necessity, sufficiency and completeness of the family.

If r=dim(U⊥), the free real symmetric block has r(r+1)/2 real parameters. Over C the Hermitian block has r real diagonal entries and two real parameters for each off-diagonal pair, giving r² real parameters. The extension is unique exactly when r=0; if r>0, adding a nonzero real multiple of the orthogonal projection onto U⊥ preserves the data. No positive-semidefinite constraint has been imposed; self-adjointness alone permits negative eigenvalues.

The two compatibility tests are independent. In the experiment x₁=e₁, x₂=e₁+e₂ and x₃=x₁+x₂. Adding e₃ only to y₃ violates the input relation but is invisible to every pairwise test because all xᵢ lie in the first coordinate plane. Conversely, changing the first coordinate of y₂ and updating y₃ to maintain the relation still defines a linear map, but generally breaks symmetry. A free complement parameter repairs neither failure.

For the compatible outputs y₁=(1,2,1), y₂=(3,4,−1), the matrix columns for e₁ and e₂ are y₁ and y₂−y₁. Thus every self-adjoint extension has rows (1,2,1), (2,2,−2), (1,−2,γ), with γ any real number. The observed plane is not invariant. In particular e₃ maps to (1,−2,γ), so it is not an eigenvector for any γ. Additional eigenvectors in the orthogonal complement require additional invariance information.

Prescribed eigenvectors are a special case

Require Tu=0 for u=(1,1,1), and Tv=2v for v=(2,−1,a). Self-adjointness forces distinct-eigenvalue vectors to be orthogonal, so u·v=1+a=0 and a=−1. At that value their span is invariant. Its perpendicular line is also invariant, and w=(0,1,−1) spans it. Every compatible self-adjoint operator is T=vvᵀ/3+γwwᵀ/2, because the squared lengths of v and w are 6 and 2. Directly check Tu=0, Tv=2v and Tw=γw. The missing eigenvalue is free, even though the missing eigenvector direction is forced.

Before consulting a solution, record the input-relation check, the pairwise symmetry check, and the free-block description separately. Then explain why the eigenvector example has an invariant complement while the general inspector does not. The spectral theorem supplies structure after compatibility is established; it does not make arbitrary prescribed data consistent.

Continue

Real eigenvalues alone do not imply self-adjointness: an upper-triangular matrix with distinct real diagonal entries may have nonorthogonal eigenspaces. The inner product and adjoint condition remain essential. Next, positive operators will use Rayleigh values to define nonnegative energy, square roots and quadratic forms. Exact positivity is a mathematical property; numerical decisions near a zero eigenvalue require tolerances and an explicit perturbation model.

Review Linear Algebra I when a prerequisite needs practice. Continue to Connect positive energy with spectral square roots.