Question textSuppose a [math: 4×4] matrix has four eigenvalues [math: −1], [math: 2], [math: 1+i] and [math: 1−i]. What is the maximum number of linearly independent eigenvectors corresponding to the eigenvalue [math: 2]?Answer: Answer 1 Question 2[input]

Question

BlackTom AI · Accepted Answer

The eigenvalue 2 appears exactly once among the four listed eigenvalues, so its algebraic multiplicity is 1.

Relevant concepts:
- Algebraic multiplicity of an eigenvalue is its multiplicity as a root of the characteristic polynomial.
- Geometric multiplicity is the dimension of the eigenspace (the number of linearly independent eigenvectors for that eigenvalue).
- For any eigenvalue, 1 ≤ geometric multiplicity ≤ algebraic multiplicity.

Step-by-step reasoning:
1. The matrix is 4×4, so the characteristic polynomial is degree 4. The listed eigenvalues −1, 2, 1 + i, and 1 − i sum (with multiplicities) to four roots, so each is given with algebraic multiplicity 1.
2. In particular, algebraic multiplicity of the eigenvalue 2 = 1.
3. By the inequality geometric multiplicity ≤ algebraic multiplicity, the geometric multiplicity of 2 is at most 1.
4. Because 2 is an eigenvalue, its eigenspace has dimension at least 1, so the geometric multiplicity is exactly 1. Thus the maximum (and actual) number of linearly independent eigenvectors for eigenvalue 2 is 1.

Why larger values are impossible:
- You cannot have 2 or more independent eigenvectors for eigenvalue 2 unless the algebraic multiplicity of 2 is ≥ 2. Here it is 1, so geometric multiplicity cannot exceed 1.
- Values like 0 are impossible for a listed eigenvalue, since being an eigenvalue guarantees at least one eigenvector.

Conclusion:
The maximum number of linearly independent eigenvectors corresponding to eigenvalue 2 is 1, which matches the provided answer.

Question textSuppose a [math: 4×4] matrix has four eigenvalues [math: −1], [math: 2], [math: 1+i] and [math: 1−i]. What is the maximum number of linearly independent eigenvectors corresponding to the eigenvalue [math: 2]?Answer: Answer 1 Question 2[input]多项填空题

类似问题

Question textSuppose a [math: 4×4] matrix has four eigenvalues [math: −1], [math: 2], [math: 1+i] and [math: 1−i]. What is the maximum number of linearly independent eigenvectors corresponding to the eigenvalue [math: 2]?Answer: Answer 1 Question 2[input]

特征向量的特征值 𝜆 是指 The eigenvalue 𝜆 of an eigen vector is

Let 𝐴 = [ 𝑎 𝑏 𝑐 𝑑 ] , where 𝑎 , 𝑏 , 𝑐 , and 𝑑 are real numbers. If [ 1 1 ] is an eigenvector of 𝐴 corresponding to the eigenvalue 𝜆 = 4 , find the value of 𝑎 + 𝑏 + 𝑐 + 𝑑 .

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