Below are five focused review questions on eigenvalues and eigenvectors. For each, follow the steps and use the tips to check your work and reasoning.

1. Compute eigenvalues and eigenvectors (2×2 upper-triangular) Matrix: A = [[4, 1], [0, 3]]

• Find eigenvalues: Compute the characteristic polynomial det(A − λI). Use the fact that for triangular matrices, eigenvalues are the diagonal entries.
• Find eigenvectors: For each eigenvalue λ, solve (A − λI)v = 0 for a nonzero vector v.
• Normalize (optional): If needed, scale eigenvectors to unit length for clarity.
• Check: Verify Av = λv for each eigenpair. Confirm that sum of eigenvalues equals tr(A) and product equals det(A).

2. Decide if a matrix is diagonalizable (geometric vs algebraic multiplicity) Matrix: B = [[2, 1, 0], [0, 2, 0], [0, 0, 3]]

• Find eigenvalues and algebraic multiplicities: Solve det(B − λI) = 0 and note repeated roots.
• Compute eigenspaces: For each eigenvalue, solve (B − λI)x = 0 and determine the dimension (geometric multiplicity).
• Conclude: B is diagonalizable iff the sum of geometric multiplicities equals 3 (or equivalently, each eigenvalue’s geometric multiplicity equals its algebraic multiplicity).
• If diagonalizable: Construct P from a basis of eigenvectors and D = diag(eigenvalues). Otherwise, explain which eigenvalue fails the criterion and why.

3. Orthogonally diagonalize a symmetric matrix Matrix: C = [[2, -1], [-1, 2]]

• Find eigenvalues: Compute det(C − λI) and solve for λ.
• Find eigenvectors: For each λ, solve (C − λI)v = 0.
• Orthonormalize: Normalize eigenvectors (symmetric matrices have orthogonal eigenvectors for distinct eigenvalues).
• Form the decomposition: Let Q be the matrix with orthonormal eigenvectors as columns; let Λ be the diagonal matrix of eigenvalues. Verify C = QΛQ^T and Q^TQ = I.

4. Reason about how eigenvalues change under matrix operations Given A ∈ R^(n×n) and λ an eigenvalue with eigenvector v ≠ 0:

• Powers: Show A^k v = λ^k v, hence eigenvalues of A^k are λ^k.
• Inverse (if A is invertible): From Av = λv with λ ≠ 0, derive A^(-1)v = (1/λ)v, so eigenvalues of A^(-1) are 1/λ.
• Shift: For α ∈ R, (A + αI)v = (λ + α)v, so eigenvalues shift by α.
• Scaling: For s ∈ R, (sA)v = (sλ)v, so eigenvalues scale by s.
• Transpose: Use det(λI − A^T) = det((λI − A)^T) = det(λI − A) to conclude A and A^T have the same eigenvalues (algebraic multiplicities match). Note: right eigenvectors of A become left eigenvectors of A^T.

5. Use eigen-decomposition to compute a matrix power Matrix: A = [[0, 1], [-2, 3]] Goal: Compute A^5 efficiently via diagonalization.

• Find eigenvalues: Solve det(A − λI) = 0.
• Find eigenvectors: For each eigenvalue λ, solve (A − λI)v = 0; assemble P from independent eigenvectors.
• Diagonalize: Verify A = PDP^(-1), with D diagonal of eigenvalues.
• Raise to a power: Compute D^5 by raising each diagonal entry to the 5th power, then A^5 = P D^5 P^(-1).
• Check: Validate by computing A^2 once directly to see the pattern, or verify A^5P = P D^5 to avoid full multiplication errors.

Tips for all problems:

• Always check Av = λv to validate eigenpairs.
• Keep track of algebraic vs geometric multiplicity; this is central to diagonalizability.
• Use trace and determinant as quick checks: sum of eigenvalues = tr(A), product = det(A) (counting algebraic multiplicities).

以下为“Python 循环与条件”复习的5个练习题。请逐题完成，按要求编写或分析代码，并参考提示巩固概念。

1. 判断分支与短路逻辑

• 任务：阅读代码，写出程序的输出，并说明进入该分支的原因。
• 代码: x = 7 if x % 2 == 0: print("A") elif 3 <= x < 10 and x % 3 == 1: print("B") else: print("C")
• 要求：解释比较链 3 <= x < 10 的含义；说明 and 的短路发生在何处。
• 提示：比较链等价于(3 <= x) and (x < 10)；and 左侧为假时右侧不计算。

2. 使用 for 与 continue 进行条件累加

• 任务：编写函数 sum_not_divisible_by_3(n)，返回1到n中所有不被3整除的整数之和。
• 要求：
  1. 使用 for 循环遍历 1..n。
  2. 使用 continue 跳过被3整除的数。
  3. 返回累加结果。
• 提示：range(1, n+1)；遇到 i % 3 == 0 时 continue。

3. 使用 while 累加直到达到阈值

• 任务：用 while 循环找出最小整数 k，使得 1+2+...+k ≥ 1000，并打印 k 与最终和。
• 要求：
  1. 使用累加器 s 与计数器 k。
  2. 循环条件基于当前和 s 是否达到阈值。
  3. 循环结束后输出结果。
• 提示：初始化 s = 0, k = 0；在循环内先更新 k，再把 k 加到 s。

4. 嵌套循环与 break：查找每行第一个负数

• 任务：给定二维列表 matrix，统计每一行中第一个负数的列索引；若该行无负数，记录为 -1。返回索引列表。
• 示例输入： matrix = [ [3, 5, -2, 7], [0, 4, 6], [1, -1, -3], [2, 8] ]
• 要求：
  1. 使用双层 for 循环。
  2. 找到第一个负数后使用 break 跳出内层循环。
  3. 每行处理结束后将结果追加到结果列表。
• 提示：为每行设置 found = -1；内层循环中一旦找到负数，记录索引并 break。

5. 理解 for-else 的执行时机

• 任务：阅读代码，分别当 n=13 与 n=15 时，写出输出并解释原因。
• 代码: n = 15 for i in range(2, int(n**0.5) + 1): if n % i == 0: print("Not prime") break else: print("Prime")
• 要求：说明 for-else 中 else 的执行前提；指出何时会跳过 else。
• 提示：for-else 的 else 在循环未遇到 break 正常结束时执行；一旦 break，else 不执行。