[THOUGHT] I'll use Python (sympy) to:

1. Define A and compute exact eigenvalues/eigenvectors.
2. Build an orthonormal eigenbasis.
3. Form projection matrices and spectral decomposition.
4. Derive a symbolic formula for A^n.
5. Compute S10, then S, then K.

I will pay attention to the definition of β and make sure I use the b3 that lies in W(λ2) as requested, and verify all arithmetic exactly. [/THOUGHT] [PYTHON] import sympy as sp

Define matrix A

A = sp.Matrix([[0,1,2], [1,0,2], [2,2,3]])

I. Eigenvalues

eigs = A.eigenvects() eigs print("Eigenvects raw:", eigs)

Extract eigenvalues sorted

eigenvals = sorted([ev[0] for ev in eigs]) print("Eigenvalues sorted:", eigenvals)

II. Eigenspaces (basis vectors)

for ev, mult, basis in eigs: print("\nEigenvalue:", ev, "mult:", mult) print("Basis for eigenspace:") for v in basis: print(sp.simplify(v))

III. Orthonormal eigenbasis

Use eigenvects and Gram-Schmidt with respect to standard inner product

First collect all eigenvectors (one from each eigenspace)

raw_vecs = [] for ev, mult, basis in eigs: # take each basis vector; likely mult=1 for all for v in basis: raw_vecs.append(sp.Matrix(v))

print("\nRaw eigenvectors:") for v in raw_vecs: print(v)

Orthonormalize

def orthonormalize(vecs): ortho = [] for v in vecs: w = v for u in ortho: w = w - (u.dot(v)) * u norm = sp.sqrt(w.dot(w)) w = sp.simplify(w / norm) ortho.append(w) return ortho

onb = orthonormalize(raw_vecs) print("\nOrthonormal eigenbasis (in the order of raw_vecs):") for v in onb: print(sp.simplify(v), "norm^2:", sp.simplify(v.dot(v)))

Associate each vector with its eigenvalue

onb_eigs = [] for v in onb: Av = A*v # solve for eigenvalue via ratio where entry nonzero lam = sp.simplify((Av[0]/v[0]) if v[0]!=0 else (Av[1]/v[1] if v[1]!=0 else Av[2]/v[2])) onb_eigs.append(lam)

print("\nEigenvalues corresponding to orthonormal basis vectors:") for lam,v in zip(onb_eigs,onb): print("lambda =", sp.simplify(lam), "vector =", sp.simplify(v))

Sort the orthonormal basis according to eigenvalues (ascending)

pairs = sorted(zip(onb_eigs,onb), key=lambda x: sp.N(x[0])) sorted_eigs = [p[0] for p in pairs] sorted_onb = [p[1] for p in pairs]

print("\nSorted eigenvalues and ON eigenvectors:") for lam,v in zip(sorted_eigs, sorted_onb): print("lambda =", sp.simplify(lam), "vector =", sp.simplify(v))

IV. Spectral decomposition: P_i = b_i b_i^T

Ps = [] for v in sorted_onb: P = v*v.T # outer product Ps.append(sp.simplify(P))

print("\nProjection matrices P_i:") for i,(lam,P) in enumerate(zip(sorted_eigs, Ps), start=1): print(f"\nP_{i}, lambda_{i}={sp.simplify(lam)}") sp.pprint(P)

Verify A = sum lambda_i P_i

A_reconstructed = sum(lam*P for lam,P in zip(sorted_eigs, Ps)) print("\nA reconstructed from spectral decomposition:") sp.pprint(sp.simplify(A_reconstructed)) print("Equals A?", sp.simplify(A_reconstructed - A) == sp.zeros(3))

V. General A^n via spectral decomposition

n = sp.symbols('n', integer=True, positive=True) A_n_symbolic = sum((lam**n)*P for lam,P in zip(sorted_eigs, Ps)) A_n_symbolic_simplified = sp.simplify(A_n_symbolic) print("\nSymbolic A^n via spectral decomposition:") sp.pprint(A_n_symbolic_simplified)

VI. Compute K

Here λ1<λ2<λ3; but problem defines λ1<λ2 and then uses W(λ2) once.

We'll interpret λ1 and λ2 as the two smallest eigenvalues.

lambda1 = sorted_eigs[0] lambda2 = sorted_eigs[1]

d1 = dim W(lambda1); eigenvalues are simple so d1=1

d1 = 1

b1,b2,b3 is orthonormal eigenbasis; assume b3 in W(lambda2).

We have sorted_onb associated with sorted_eigs; pick basis accordingly:

b1, b2, b3 = sorted_onb # b1->lambda1, b2->lambda2, b3->lambda3

But problem wants b3 in W(lambda2), so we should reorder basis so that

the vector for lambda2 is placed as b3.

Let's explicitly assign:

eig_to_vec = {sorted_eigs[i]: sorted_onb[i] for i in range(3)} b_lambda1 = eig_to_vec[lambda1] b_lambda2 = eig_to_vec[lambda2] b_lambda3 = eig_to_vec[sorted_eigs[2]]

We must now define (b3) specifically as belonging to W(lambda2),

so let b3 = b_lambda2

b3_vec = b_lambda2

beta = |(b3)_1|^2

beta = sp.simplify(abs(b3_vec[0])**2) print("\nChosen b3 (eigenvector for lambda2):", b3_vec) print("beta =", beta)

Compute A^10 and S10

A10 = A**10 S10 = sp.simplify(sum(A10[i,j] for i in range(3) for j in range(3))) print("\nA^10 =") sp.pprint(A10) print("S10 =", S10)

S = sp.simplify((lambda2 - lambda1)d1beta**(-1)*S10) print("S =", S)

K = sp.simplify(999*S**2 + 7) print("K =", K) [/PYTHON]

[THOUGHT] \boxed{3512109464910000575431}