Let \(A\) be \(n\times n\) real valued square symmetric matrix of rank 2 with…
2017
Let \(A\) be \(n\times n\) real valued square symmetric matrix of rank 2 with \(\sum_{i=1}^{n}\sum_{j=1}^{n}A^{2}_{ij} = 50.\). Consider the following statements.
I. One eigenvalue must be in [−5,5]
II. The eigenvalue with the largest magnitude must be strictly greater than 5
Which of the above statements about eigenvalues of \(A\) is/are necessarily CORRECT?
- A.
Both I and II
- B.
I only
- C.
II only
- D.
Neither I nor II
Attempted by 66 students.
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Correct answer: B
Key facts: a real symmetric matrix has real eigenvalues and is orthogonally diagonalizable. The Frobenius norm squared (sum of squares of all entries) equals the sum of the squares of all eigenvalues.
Rank 2 means at most two nonzero eigenvalues; denote them by λ1 and λ2 (all other eigenvalues are 0).
Given sum of squares of entries = 50, we have λ1² + λ2² = 50.
Analysis of the first statement: If both nonzero eigenvalues had absolute value greater than 5 then λ1² > 25 and λ2² > 25, so λ1² + λ2² > 50, contradicting λ1² + λ2² = 50. Hence at least one eigenvalue satisfies |λ| ≤ 5, i.e. it lies in [-5,5]. So the first statement is necessarily true.
Analysis of the second statement: The largest eigenvalue magnitude need not be strictly greater than 5. For example, take the two nonzero eigenvalues λ1 = 5 and λ2 = 5 (all others zero). Then λ1² + λ2² = 25 + 25 = 50, but the largest magnitude is 5, not greater than 5. Thus the second statement is not necessarily true.
Conclusion: Only the first statement is necessarily correct.