GATE CS 2018 Q.27 (CS Q.17): Linear Algebra

Consider a matrix $A = u v^T$ where $u = \begin{pmatrix} 1 \\ 2 \end{pmatrix}, v = \begin{pmatrix} 1 \\ 1 \end{pmatrix}$ . Note that $v^T$ denotes the transpose of $v$ . The largest eigenvalue of $A$ is _____.

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Official answer and solution

Official answer: 3
From: Official GATE 2018 answer key

Solution

Recognize that the outer product of two vectors produces a rank-1 matrix whose non-zero eigenvalue equals the inner product $v^T u$ .

Matrix $A = u v^T$ is the outer product of column vector $u = \begin{pmatrix} 1 \\ 2 \end{pmatrix}$ and row vector $v^T = \begin{pmatrix} 1 & 1 \end{pmatrix}$ :

A = \begin{pmatrix} 1 \\ 2 \end{pmatrix} \begin{pmatrix} 1 & 1 \end{pmatrix} = \begin{pmatrix} 1 & 1 \\ 2 & 2 \end{pmatrix}

Since $A$ is a rank-1 matrix of size $2 \times 2$ , one eigenvalue is 0.

The trace of $A$ is the sum of its diagonal entries:

\text{tr}(A) = 1 + 2 = 3

Alternatively, for $A = u v^T$ , the non-zero eigenvalue is given by the inner product:

\lambda = v^T u = 1 \times 1 + 1 \times 2 = 3

The eigenvalues of $A$ are 3 and 0. Therefore, the largest eigenvalue is 3.

Final answer

The answer is the one in the official GATE answer key. The solution is this site's own explanation.

Source

Paper: GATE 2018 Computer Science and Information Technology (CS)
Question: 27 of 65, printed as CS Q.17
Section: Computer Science and Information Technology

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