Problem Analysis #7

34.81s

10,354 toks

Problem Statement

Answer the following questions about a real symmetric matrix $A$ : $A =\begin{pmatrix}0 & 1 & 2 \\1 & 0 & 2 \\2 & 2 & 3\end{pmatrix}.$

[I.] Find all the different eigenvalues of matrix $A$ , $\lambda_1, \cdots, \lambda_r$ ( $\lambda_1 < \cdots < \lambda_r$ ).
[II.] Find all the eigenspaces $W(\lambda_1), \cdots, W(\lambda_r)$ corresponding to $\lambda_1, \cdots, \lambda_r$ , respectively.
[III.] Find an orthonormal basis $b_1, b_2, b_3$ consisting of eigenvectors of $A$ , each belonging to one of $W(\lambda_1), \cdots, W(\lambda_r)$ obtained in Question II.
[IV.] Find the spectral decomposition of $A$ : $A = \sum_{i=1}^{r} \lambda_i P_i,$ where $P_i$ is the projection matrix onto $W(\lambda_i)$ .
[V.] Find $A^n$ , where $n$ is any positive integer.
[VI.] Let $\lambda_1 < \lambda_2$ be the eigenvalues from {\rm I}, and let $d_1 := \dim W(\lambda_1).$ Let $b_1,b_2,b_3$ be the orthonormal eigenbasis from {\rm III}, and assume $b_3 \in W(\lambda_2)$ . Denote by $(b_3)_1$ the first component of $b_3$ , and set $\beta := |(b_3)_1|^2.$ For $n=10$ , let $S_{10}$ be the sum of all entries of the matrix $A^{10}$ : $S_{10} := \sum_{i=1}^3 \sum_{j=1}^3 (A^{10})_{ij}.$ Define $S := (\lambda_2 - \lambda_1)\, d_1\, \beta^{-1} S_{10}, \qquad \mathcal{K} := 999\,S^2 + 7.$ Compute the integer $\mathcal{K}$ . \

Ground Truth

3512109464910000575431

Execution Trace

Reasoning Process

[THOUGHT] Let me solve this step by step. I'll use Python to compute all the required values.

First, I need to:

Find eigenvalues of matrix A
Find eigenspaces
Find orthonormal eigenbasis
Find spectral decomposition
Find A^n
Compute the final value K

Let me start by computing the eigenvalues and eigenvectors. [/THOUGHT] [PYTHON] import numpy as np from numpy import linalg as LA from fractions import Fraction import sympy as sp

Define the matrix A

A = np.array([[0, 1, 2], [1, 0, 2], [2, 2, 3]], dtype=float)

Find eigenvalues and eigenvectors

eigenvalues, eigenvectors = LA.eig(A)

Sort eigenvalues and corresponding eigenvectors

idx = np.argsort(eigenvalues) eigenvalues = eigenvalues[idx] eigenvectors = eigenvectors[:, idx]

print("Eigenvalues (sorted):", eigenvalues) print("\nEigenvectors (columns):") print(eigenvectors)

Let's also compute eigenvalues symbolically using sympy

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

Characteristic polynomial

lam = sp.Symbol('lambda') char_poly = A_sym.charpoly(lam) print("\nCharacteristic polynomial:", char_poly.as_expr())

Find eigenvalues symbolically

eigenvalues_sym = A_sym.eigenvals() print("\nSymbolic eigenvalues:", eigenvalues_sym)

Get eigenvectors

eigenvects_sym = A_sym.eigenvects() print("\nSymbolic eigenvectors:") for ev in eigenvects_sym: print(f"Eigenvalue {ev[0]}, multiplicity {ev[1]}, eigenvectors: {ev[2]}") [/PYTHON]

Tool Output

[PYTHON OUTPUT] Eigenvalues (sorted): [-1. -1. 5.] Eigenvectors (columns): [[-0.91287093 -0.27590253 0.40824829] [ 0.18257419 -0.7974171 0.40824829] [ 0.36514837 0.53665982 0.81649658]] Characteristic polynomial: lambda**3 - 3*lambda**2 - 9*lambda - 5 Symbolic eigenvalues: {5: 1, -1: 2} Symbolic eigenvectors: Eigenvalue -1, multiplicity 2, eigenvectors: [Matrix([ [-1], [ 1], [ 0]]), Matrix([ [-2], [ 0], [ 1]])] Eigenvalue 5, multiplicity 1, eigenvectors: [Matrix([ [1/2], [1/2], [ 1]])] [/PYTHON OUTPUT]

Reasoning Process

[THOUGHT] Let me continue computing the required values for part VI.

From the output:

λ₁ = -1 (with multiplicity 2)
λ₂ = 5 (with multiplicity 1)
d₁ = dim W(λ₁) = 2

The eigenvector for λ₂ = 5 is [1/2, 1/2, 1]ᵀ. To normalize it:

Norm = √((1/2)² + (1/2)² + 1²) = √(1/4 + 1/4 + 1) = √(3/2) = √6/2
Normalized b₃ = [1/2, 1/2, 1]ᵀ / (√6/2) = [1/√6, 1/√6, 2/√6]ᵀ

So (b₃)₁ = 1/√6, and β = |(b₃)₁|² = 1/6

For A^10: A^10 = λ₁^10 · P₁ + λ₂^10 · P₂ = (-1)^10 · P₁ + 5^10 · P₂ = P₁ + 5^10 · P₂

Since P₁ + P₂ = I (identity), we have P₁ = I - P₂

P₂ = b₃ · b₃ᵀ where b₃ = [1/√6, 1/√6, 2/√6]ᵀ

The sum of all entries of P₂: P₂ has entries (b₃)ᵢ(b₃)ⱼ, so sum = (Σᵢ(b₃)ᵢ)² = (1/√6 + 1/√6 + 2/√6)² = (4/√6)² = 16/6 = 8/3

Sum of all entries of I = 3 (trace of 3×3 identity) Sum of all entries of P₁ = 3 - 8/3 = 1/3

S₁₀ = sum of entries of A^10 = 1·(1/3) + 5^10·(8/3) = (1 + 5^10·8)/3

5^10 = 9765625 S₁₀ = (1 + 9765625·8)/3 = (1 + 78125000)/3 = 78125001/3 = 26041667

Now: S = (λ₂ - λ₁)·d₁·β⁻¹·S₁₀ = (5-(-1))·2·6·26041667 = 6·2·6·26041667 = 72·26041667 = 1875000024

K = 999·S² + 7 = 999·(1875000024)² + 7

Let me verify this calculation: S = 72 × 26041667 = 1875000024 S² = 1875000024² = 3515625090000000576 K = 999 × 3515625090000000576 + 7 = 3512109464910000575424 + 7 = 3512109464910000575431

Wait, let me recalculate more carefully with Python. [/THOUGHT] [PYTHON] import numpy as np from fractions import Fraction

Given values from symbolic computation

lambda1 = -1 lambda2 = 5 d1 = 2 # dimension of W(lambda1)

Eigenvector for lambda2 = 5 is [1/2, 1/2, 1]

Normalize it

v = np.array([Fraction(1,2), Fraction(1,2), Fraction(1,1)]) norm_sq = sum(x**2 for x in v) print(f"Norm squared of eigenvector: {norm_sq}")

Normalized eigenvector b3

b3 = v / sqrt(norm_sq) = v / sqrt(3/2)

(b3)_1 = (1/2) / sqrt(3/2) = (1/2) * sqrt(2/3) = 1/sqrt(6)

beta = |(b3)_1|^2 = 1/6

beta = Fraction(1, 6) print(f"beta = {beta}")

A^10 = lambda1^10 * P1 + lambda2^10 * P2

P2 = b3 * b3^T (projection onto eigenspace of lambda2)

Sum of entries of P2 = (sum of components of b3)^2

sum of components of b3 = (1/2 + 1/2 + 1) / sqrt(3/2) = 2 / sqrt(3/2) = 2 * sqrt(2/3) = 2sqrt(2)/sqrt(3) = 2sqrt(6)/3

Actually: sum = (1/2 + 1/2 + 1) / sqrt(3/2) = 2 * sqrt(2/3) = 2*sqrt(2)/sqrt(3)

(sum)^2 = 4 * (2/3) = 8/3

sum_P2 = Fraction(8, 3) sum_I = 3 # sum of all entries of identity matrix sum_P1 = sum_I - sum_P2 print(f"Sum of entries of P1: {sum_P1}") print(f"Sum of entries of P2: {sum_P2}")

A^10 = (-1)^10 * P1 + 5^10 * P2 = P1 + 5^10 * P2

lambda1_10 = (-1)10 lambda2_10 = 510 print(f"5^10 = {lambda2_10}")

S10 = lambda1_10 * sum_P1 + lambda2_10 * sum_P2 print(f"S10 = {S10}")

S = (lambda2 - lambda1) * d1 * beta^(-1) * S10

S = (lambda2 - lambda1) * d1 * (1/beta) * S10 print(f"S = {S}")

K = 999 * S^2 + 7

K = 999 * S**2 + 7 print(f"K = {K}") [/PYTHON]

Tool Output

[PYTHON OUTPUT] Norm squared of eigenvector: 3/2 beta = 1/6 Sum of entries of P1: 1/3 Sum of entries of P2: 8/3 5^10 = 9765625 S10 = 26041667 S = 1875000024 K = 3512109464910000575431 [/PYTHON OUTPUT]

Reasoning Process

[THOUGHT] The Python calculation confirms all the steps:

λ₁ = -1, λ₂ = 5
d₁ = 2 (dimension of eigenspace for λ₁)
β = 1/6 (square of first component of normalized eigenvector b₃)
S₁₀ = 26041667 (sum of all entries of A¹⁰)
S = (λ₂ - λ₁) · d₁ · β⁻¹ · S₁₀ = 6 · 2 · 6 · 26041667 = 1875000024
K = 999 · S² + 7 = 3512109464910000575431 [/THOUGHT]

\boxed{3512109464910000575431}

Final Model Output

3512109464910000575431

Comparative Analysis (P7) (Average across runs)

Avg Time Taken

Deepseek-V3.2

716.8

Gemini-3-Pro-Preview

238.5

Llama-VARCO-8B-Instruct

171.0

GPT-5.1 (high)

91.4

Grok-4.1-fast

70.1

Solar-Pro-2 (31B)(high)

68.0

Claude-Opus-4.5

34.8

A.X-4.0 (72B)

32.0

HCX-007(high)

29.5

EXAONE-4.0.1-32B (high)

2.2

Avg Token Usage

Grok-4.1-fast

14838.0

Deepseek-V3.2

14663.0

Llama-VARCO-8B-Instruct

13773.0

Solar-Pro-2 (31B)(high)

12124.0

GPT-5.1 (high)

11204.0

Gemini-3-Pro-Preview

10464.0

Claude-Opus-4.5

10354.0

A.X-4.0 (72B)

5137.0

HCX-007(high)

4970.0

EXAONE-4.0.1-32B (high)

3503.0