Singular Value Decomposition and Principal Component Analysis Previous Year Questions (PYQs) for GATE DA: 2+ Solved Questions with Step-by-Step Solutions

Solve 2+ Singular Value Decomposition and Principal Component Analysis previous year questions for GATE DA with answers and detailed solutions. Free sample questions below.

Chapter Roadmap: SVD and PCA

Orientation

Chapter Roadmap: SVD and PCA

Matrix Decompositions, with the current stop on PCA eigenvalues and the direction of maximum variance.

Current topic: PCA Eigenvalues and Maximum Variance Direction

Core idea: covariance eigenvalues measure variance along principal directions.

Goal: identify the direction that captures maximum variance.

Exam focus: direct use of the largest eigenvalue and the quadratic form.

Next topic: Rank-One Matrices and Singular Values

Core idea: outer products create simple matrices with clean spectral structure.

Goal: connect singular values to rank-one constructions.

Exam focus: singular values of structured matrices.

The Goal: Finding the Direction of Maximum Spread

Concept

The Goal: Finding the Direction of Maximum Spread

Geometric Meaning

A centered data cloud can be spread differently along different directions. Principal Component Analysis asks:

Which unit direction keeps the largest spread when all points are projected onto it?

That direction is called the first principal component.

Algebraic Translation

If $Σ$ is the covariance matrix of centered data, then the best direction is an eigenvector of $Σ$ .

Σ v = λ v

The direction with the largest variance is the eigenvector corresponding to

λ_{m a x}

Key Mapping

Geometry	Linear Algebra
Direction of projection	Unit vector $u$
Spread along that direction	$u^{T} Σ u$
Best direction	Eigenvector for $λ_{m a x}$
Variance kept there	$λ_{m a x}$

Singular Value Decomposition and Principal Component Analysis: Solved Questions with Step-by-Step Explanations (2 Problems)

Question 1 · Linear Algebra NAT

Let

D = {x^{(1)}, ..., x^{(n)}}

be a dataset of

n

observations where each

x^{(i)} \in R^{100}

. It is given that

\sum_{i = 1}^{n} x^{(i)} = 0

. The covariance matrix computed from

D

has eigenvalues

λ_{i} = 100 2^{- i}

1 \leq i \leq 100

. Let

u \in R^{100}

be the direction of maximum variance with

u^{T} u = 1

.

The value of

\frac{1}{n} i = 1 \sum n (u^{T} x^{(i)})^{2} =

(Answer in integer)

Correct Answer:

Step-by-Step Solution

Key idea: This is a PCA maximum variance direction problem. The expression given is the definition of variance along a direction, which is maximized by the largest eigenvalue of the covariance matrix.

Step 1: Recognize that the dataset has mean zero ( $\sum_{i = 1}^{n} x^{(i)} = 0$ ). Thus, the covariance matrix $C$ is given by $C = \frac{1}{n} \sum_{i = 1}^{n} x^{(i)} (x^{(i)})^{T}$ .

Step 2: The expression to evaluate is $\frac{1}{n} \sum_{i = 1}^{n} (u^{T} x^{(i)})^{2}$ . Notice that $(u^{T} x^{(i)})^{2} = (u^{T} x^{(i)}) (x^{(i) T} u) = u^{T} (x^{(i)} (x^{(i)})^{T}) u$ .

Step 3: Substitute this into the sum: $\frac{1}{n} \sum_{i = 1}^{n} u^{T} (x^{(i)} (x^{(i)})^{T}) u = u^{T} (\frac{1}{n} \sum_{i = 1}^{n} x^{(i)} (x^{(i)})^{T}) u = u^{T} C u$ .

Step 4: The problem states that $u$ is the direction of maximum variance with $u^{T} u = 1$ . By the properties of PCA, the maximum variance along any unit vector is the largest eigenvalue of the covariance matrix $C$ .

Step 5: The eigenvalues are given as $λ_{i} = 100 \cdot 2^{- i}$ for $1 \leq i \leq 100$ . The largest eigenvalue occurs at $i = 1$ , which is $λ_{1} = 100 \cdot 2^{- 1} = 50$ .

Step 6: Therefore, $u^{T} C u = λ_{1} = 50$ .

Answer: 50

Question 2 · Linear Algebra NAT

Let

u = 12345

, and let

σ_{1}, σ_{2}, σ_{3}, σ_{4}, σ_{5}

be the singular values of the matrix

M = u u^{T}

(where

u^{T}

is the transpose of

u

). The value of

\sum_{i = 1}^{5} σ_{i}

is ______.

Correct Answer:

Step-by-Step Solution

Key idea: This is a rank-one matrix singular value problem. The matrix is given as an outer product $M = u u^{T}$ .

Step 1: Recognize that $M = u u^{T}$ is a symmetric, positive semi-definite matrix of rank 1.

Step 2: For a rank-one matrix of the form $u u^{T}$ , the only non-zero eigenvalue is given by $u^{T} u = ∥ u ∥^{2}$ .

Step 3: Calculate the squared Euclidean norm: $∥ u ∥^{2} = 1^{2} + 2^{2} + 3^{2} + 4^{2} + 5^{2} = 1 + 4 + 9 + 16 + 25 = 55$ .

Step 4: Since $M$ is symmetric and positive semi-definite, its singular values are the absolute values of its eigenvalues. Thus, the singular values are $σ_{1} = 55$ and $σ_{2} = σ_{3} = σ_{4} = σ_{5} = 0$ .

Step 5: The sum of the singular values is $55 + 0 + 0 + 0 + 0 = 55$ .

Answer: 55

A question from this chapter

Question 1

Let D = {x^{(1)}, ..., x^{(n)}} be a dataset of n observations where each x^{(i)} \in R^{100} . It is given that \sum_{i = 1}^{n} x^{(i)} = 0 . The covariance matrix computed from D has eigenvalues λ_{i} = 100 2^{- i}, 1 \leq i \leq 100 . Let u \in R^{100} be the direction of maximum variance with u^{T} u = 1 . The value of \frac{1}{n} i = 1 \sum n (u^{T} x^{(i)})^{2} = (Answer in integer)

Question 2

Let u = 12345, and let σ_{1}, σ_{2}, σ_{3}, σ_{4}, σ_{5} be the singular values of the matrix M = u u^{T} (where u^{T} is the transpose of u). The value of \sum_{i = 1}^{5} σ_{i} is ______.

Free preview ends here

Login to view the complete previous-year questions and solutions

Creating an account is free. You get the rest of this chapter, step-by-step solutions, and a study plan built around the topics you are actually weak at.

Why MastersUp

Personalised first. High quality throughout.

Most platforms hand everyone the same content. Here the content moves with your performance, topic by topic.

Built around you, not around a syllabus PDF

Every answer you give moves your topic-level intelligence rate. The next question, the next revision card and tomorrow's plan all change with it.

Revision that hits your weak spots

We only revise topics you have actually attempted and are still below the safe bar on — never the same chapter on repeat.

Questions calibrated to the real exam

Each question carries a measured toughness. You are served a rung above your current level, so practice keeps stretching you.

Notes written for recall, not for volume

Full lesson cards for first study, curated short-note cards for the last mile — with derivations, traps and exam patterns marked.

One place for everything

Notes, chapter practice, previous-year questions, test series and full-length papers — all feeding one picture of your preparation.

Honest progress

No vanity streaks. Progress here means chapters mastered and accuracy that held up on harder questions.

Unlock the whole course

Full notes and short notes, the complete question bank with worked solutions, mock tests, full-length papers, and an adaptive plan that rebuilds itself as you improve.

View all plans Start free

Singular Value Decomposition and Principal Component Analysis Previous Year Questions (PYQs) for GATE DA: 2+ Solved Questions with Step-by-Step Solutions

Chapter Roadmap: SVD and PCA

Chapter Roadmap: SVD and PCA

Current topic: PCA Eigenvalues and Maximum Variance Direction

Next topic: Rank-One Matrices and Singular Values

The Goal: Finding the Direction of Maximum Spread

The Goal: Finding the Direction of Maximum Spread

Geometric Meaning

Algebraic Translation

Key Mapping

Singular Value Decomposition and Principal Component Analysis: Solved Questions with Step-by-Step Explanations (2 Problems)

Step-by-Step Solution

Step-by-Step Solution

More previous year questions (pyqs) in this unit

A question from this chapter

Login to view the complete previous-year questions and solutions

Personalised first. High quality throughout.

Built around you, not around a syllabus PDF

Revision that hits your weak spots

Questions calibrated to the real exam

Notes written for recall, not for volume

One place for everything

Honest progress

Unlock the whole course