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Matrix Decompositions Practice Questions 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

Level 3: Exam Standard

A centered dataset of n = 50 observations in R^{3} has scatter matrix S = \sum_{i = 1}^{50} x^{(i)} (x^{(i)})^{T} with eigenvalues 735, 245, 0 . The covariance matrix is defined as Σ = \frac{1}{n} S . A student claims: "The largest eigenvalue of Σ is 15, and the sum of all eigenvalues of Σ is 20 ." These claims are inconsistent with Σ = \frac{1}{n} S . Find the minimum integer k such that the sum of the k largest eigenvalues of the true Σ is at least 15 .

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Matrix Decompositions Practice Questions 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 Practice Questions

104 practice questions for Matrix Decompositions in GATE DA, sorted chapter by chapter and graded from basic to exam level, each with a full solution.

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 MCQ

A centered dataset of $n = 50$ observations in $R^{3}$ has scatter matrix $S = \sum_{i = 1}^{50} x^{(i)} (x^{(i)})^{T}$ with eigenvalues $735, 245, 0$ . The covariance matrix is defined as $Σ = \frac{1}{n} S$ .

A student claims: "The largest eigenvalue of $Σ$ is $15$ , and the sum of all eigenvalues of $Σ$ is $20$ ."

These claims are inconsistent with $Σ = \frac{1}{n} S$ . Find the minimum integer $k$ such that the sum of the $k$ largest eigenvalues of the true $Σ$ is at least $15$ .

A.
$1$
B.
$0$
C.
$2$
D.
$3$

Correct Answer:

Step-by-Step Solution

Key idea: this is a contradiction problem where the student's claims reveal a unit mismatch ( $n$ vs $n - 1$ ), and you must compute the true eigenvalues of $Σ$ to answer the minimum $k$ question.

Step 1: compute the true eigenvalues of $Σ = \frac{1}{50} S$ .

The eigenvalues of $S$ are $735, 245, 0$ . So the eigenvalues of $Σ$ are:

$λ_{1} = 735/50 = 14.7$

$λ_{2} = 245/50 = 4.9$

$λ_{3} = 0/50 = 0$

Step 2: verify the student's claims are contradictory.

The student claims $λ_{m a x} = 15$ and sum $= 20$ .

True $λ_{m a x} = 14.7 \neq = 15$ . True sum $= 14.7 + 4.9 + 0 = 19.6 \neq = 20$ .

(The student likely used $\frac{1}{n - 1} S = \frac{1}{49} S$ , giving eigenvalues $15, 5, 0$ with sum $20$ .)

Step 3: find the minimum $k$ such that the sum of the $k$ largest eigenvalues of $Σ$ is at least $15$ .

$k = 1$ : sum $= 14.7 < 15$ .

$k = 2$ : sum $= 14.7 + 4.9 = 19.6 \geq 15$ .

So the minimum $k$ is $2$ .

Answer: $2$

A question from this chapter

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Matrix Decompositions Practice Questions for GATE DA

About Matrix Decompositions Practice Questions

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