GATE 2026 DA Q.11: solution and answer

Official answer: BSource: Official answer key, GATE 2026 DA (IIT Guwahati)

1. The principal components of the 100 features are the unit eigenvectors of the covariance matrix $\Sigma$ (real and symmetric), ordered by decreasing eigenvalue: ${\mathbf{v}}_1$ has the largest eigenvalue and ${\mathbf{v}}_{10}$ the tenth largest.

2. A real symmetric matrix has an orthonormal set of eigenvectors (spectral theorem). Directly: if $\Sigma {\mathbf{v}}_1={\lambda }_1{\mathbf{v}}_1$ and $\Sigma {\mathbf{v}}_{10}={\lambda }_{10}{\mathbf{v}}_{10}$ with ${\lambda }_1\ne {\lambda }_{10}$, then $${\lambda }_1{\mathbf{v}}_1^{\top }{\mathbf{v}}_{10}=(\Sigma {\mathbf{v}}_1{)}^{\top }{\mathbf{v}}_{10}={\mathbf{v}}_1^{\top }{\Sigma }^{\top }{\mathbf{v}}_{10}={\mathbf{v}}_1^{\top }\Sigma {\mathbf{v}}_{10}={\lambda }_{10}{\mathbf{v}}_1^{\top }{\mathbf{v}}_{10},$$ so $({\lambda }_1-{\lambda }_{10}){\mathbf{v}}_1^{\top }{\mathbf{v}}_{10}=0$, which gives ${\mathbf{v}}_1^{\top }{\mathbf{v}}_{10}=0$.

3. If some eigenvalues are repeated, the eigenvectors within a repeated eigenspace can still be chosen orthonormal, and PCA does choose them so. Hence the principal components are always mutually orthogonal.

4. Therefore $\cos \theta =\frac{{\mathbf{v}}_1^{\top }{\mathbf{v}}_{10}}{\Vert {\mathbf{v}}_1\Vert \Vert {\mathbf{v}}_{10}\Vert }=0$, so $\theta =90^{\circ }$.

5. Reducing the dimension from 100 to 10 only keeps the top 10 components and discards the other 90. It does not change the directions of the kept components, so they remain orthogonal and $\theta =90^{\circ }$. This is option (B); (A), (C) and (D) all require a non-zero dot product or parallel vectors.