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Matrix Decompositions Notes 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 Notes 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 Notes

Full study notes for Matrix Decompositions in GATE DA, organised across 4 chapters. Each chapter page explains concepts from the basics with worked examples and the formulas you need.

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 Notes for GATE DA

About Matrix Decompositions Notes

Matrix Decompositions Weightage in GATE DA

Matrix Decompositions Chapter Matrix

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One Solved Question from Each Matrix Decompositions Chapter

Step-by-Step Solution