GATE CS 2017 Set 1 Q31 — Eigenvalues and Eigenvectors, LU Decomposition

For any real symmetric matrix $A$, the sum of the squares of its elements is equal to the sum of the squares of its eigenvalues (this is the square of the Frobenius norm, denoted as $\|A\|_F^2$):

Since the rank of $A$ is $2$, there are exactly two non-zero eigenvalues, say $\lambda_1$ and $\lambda_2$. All other $n-2$ eigenvalues are $0$. Therefore, we have:

Let us evaluate the given statements:

Statement I: One eigenvalue must be in $[-5, 5]$

Suppose, for contradiction, that no eigenvalue lies in $[-5, 5]$. Since all other $n-2$ eigenvalues are $0$ (which lies in $[-5, 5]$), this assumption requires both non-zero eigenvalues to satisfy:

Squaring both inequalities:

Adding them gives:

This directly contradicts our condition $\lambda_1^2 + \lambda_2^2 = 50$. Thus, at least one eigenvalue must lie in the interval $[-5, 5]$.
Statement I is necessarily CORRECT.

Statement II: The eigenvalue with the largest magnitude must be strictly greater than $5$

Consider the case where $\lambda_1 = 5$ and $\lambda_2 = -5$. Here:

The magnitude of both eigenvalues is $5$. The largest magnitude is $5$, which is not strictly greater than $5$. Since we found a valid counterexample, Statement II is not necessarily correct.

Thus, only Statement I is necessarily correct.