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Matrix Decompositions PYQs for GATE DA

GATE DA Matrix Decompositions: 4 chapters, 2 previous year questions (10% of Linear Algebra), 104 practice questions and one solved question from each chapter

A question from this chapter

Question 1

2025 PYQ

Level 3: Exam Standard

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)

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Matrix Decompositions PYQs for GATE DA

GATE DA Matrix Decompositions: 4 chapters, 2 previous year questions (10% of Linear Algebra), 104 practice questions and one solved question from each chapter.

About Matrix Decompositions Previous Year Questions (PYQs)

2 previous year questions from Matrix Decompositions in GATE DA, grouped by chapter with the exam year, answer key and step-by-step solution for each.

Matrix Decompositions Weightage in GATE DA

Matrix Decompositions accounts for 2 of 20 Linear Algebra previous year questions in our bank (10%), about 1 per paper across 2 papers.

Matrix Decompositions Chapter Matrix

Chapter	Topics	PYQs	Share of unit PYQs	Practice questions
Chapter 1 — Matrix Decompositions		0	0%	0
Singular Value Decomposition and Principal Component Analysis	PCA Eigenvalues and Maximum Variance Direction, Rank-One Matrices and Singular Values	2	100%	104
Chapter 2 — Matrix Decompositions		0	0%	0
Chapter 3 — Matrix Decompositions		0	0%	0

One Solved Question from Each Matrix Decompositions Chapter

Question 1 · Singular Value Decomposition and Principal Component Analysis · 2025 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

A question from this chapter

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Matrix Decompositions PYQs for GATE DA

About Matrix Decompositions Previous Year Questions (PYQs)

Matrix Decompositions Weightage in GATE DA

Matrix Decompositions Chapter Matrix

More from Linear Algebra

One Solved Question from Each Matrix Decompositions Chapter

Step-by-Step Solution