Singular Value Decomposition and Principal Component Analysis Practice Questions for GATE DA: 15+ Solved Questions with Step-by-Step Solutions

Solve 15+ Singular Value Decomposition and Principal Component Analysis practice 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 (5 Problems)

Question 1 · Linear Algebra MCQ

In Principal Component Analysis (PCA), the first principal component is defined as the direction that:

A.
Minimizes the reconstruction error of the data
B.
Maximizes the variance of the projected data
C.
Minimizes the trace of the covariance matrix
D.
Maximizes the determinant of the covariance matrix

Correct Answer:

Step-by-Step Solution

Key idea: This is a definition-based question about the primary objective of PCA.

Step 1: Recall that PCA seeks to find orthogonal directions (principal components) that capture the most information in the data.

Step 2: Information in this context is measured by variance. The first principal component is specifically the unit vector $u$ that maximizes the variance of the data when projected onto it.

Answer: Maximizes the variance of the projected data.

Question 2 · Linear Algebra MCQ

If $v$ is an eigenvector of the covariance matrix $C$ with corresponding eigenvalue $λ$ , what is the variance of the data along the direction $v$ (assuming $v$ is normalized)?

A.
$λ^{2}$
B.
$λ$
C.
$λ$
D.
$1/ λ$

Correct Answer:

Step-by-Step Solution

Key idea: This question connects the spectral decomposition of the covariance matrix to statistical variance.

Step 1: The variance along a direction $v$ is $v^{T} C v$ .

Step 2: Since $v$ is an eigenvector, $C v = λ v$ .

Step 3: Substitute this into the variance formula: $v^{T} (λ v) = λ (v^{T} v)$ .

Step 4: Since $v$ is normalized, $v^{T} v = 1$ . Thus, the variance is $λ$ .

Answer: $λ$

Question 3 · Linear Algebra MCQ

In Principal Component Analysis, if a unit vector $u$ is constrained to be orthogonal to the first principal component, what is the maximum possible variance of the data projected onto $u$ ?

A.
The largest eigenvalue of the covariance matrix
B.
The sum of all eigenvalues
C.
The second largest eigenvalue of the covariance matrix
D.
Zero

Correct Answer:

Step-by-Step Solution

Key idea: This is a comparison question, recognizable because it asks for the maximum variance under an orthogonality constraint to the first PC.

Why it applies: The variational characterization of eigenvalues (Courant-Fischer theorem) states that the maximum variance in the subspace orthogonal to the first $k - 1$ principal components is the $k$ -th largest eigenvalue.

Step 1: The first principal component corresponds to the largest eigenvalue $λ_{1}$ .

Step 2: We are looking for the maximum variance in the subspace orthogonal to this first component.

Step 3: By definition of PCA, the direction that maximizes variance in this orthogonal subspace is the second principal component.

Step 4: The variance along the second principal component is the second largest eigenvalue, $λ_{2}$ .

Answer: The correct option is C.

Trap: Assuming the maximum variance is still the largest eigenvalue, ignoring the orthogonality constraint.

Question 4 · Linear Algebra MCQ

Let $C$ be the covariance matrix of a dataset, and let $w$ be a unit vector. Which expression represents the variance of the data when projected onto the direction $w$ ?

A.
$w^{T} C w$
B.
$w^{T} C^{- 1} w$
C.
$tr (C w)$
D.
$det (C)$

Correct Answer:

Step-by-Step Solution

Key idea: This question tests the algebraic formula for projected variance using the covariance matrix.

Step 1: The projection of a data point $x$ onto a unit vector $w$ is the scalar $w^{T} x$ .

Step 2: The variance of these projections is given by the quadratic form $w^{T} C w$ .

Answer: $w^{T} C w$

Question 5 · Linear Algebra MCQ

A centered dataset in $R^{d}$ has a covariance matrix with eigenvalues given by $λ_{i} = 80 \cdot (1/2)^{i}$ for $i = 1, 2, \dots, d$ . What is the maximum possible variance of the data when projected onto any single unit direction?

A.
80
B.
20
C.
40
D.
10

Correct Answer:

Step-by-Step Solution

Key idea: This is a sequence evaluation question, recognizable because it gives a formula for eigenvalues and asks for the maximum variance.

Why it applies: The maximum variance in any direction is the largest eigenvalue of the covariance matrix.

Step 1: The eigenvalues are given by $λ_{i} = 80 \cdot (1/2)^{i}$ for $i = 1, 2, \dots, d$ .

Step 2: Since $(1/2)^{i}$ is a decreasing sequence, the largest eigenvalue occurs at the smallest index, $i = 1$ .

Step 3: Calculate $λ_{1} = 80 \cdot (1/2)^{1} = 40$ .

Step 4: The maximum possible variance of the data when projected onto any single unit direction is exactly this largest eigenvalue.

Answer: The correct option is C.

Trap: Evaluating the sequence at $i = 0$ (which would give 80) or summing the series. The problem explicitly specifies $i \geq 1$ .

A question from this chapter

Question 1

In Principal Component Analysis (PCA), the first principal component is defined as the direction that:

Question 2

If v is an eigenvector of the covariance matrix C with corresponding eigenvalue λ, what is the variance of the data along the direction v (assuming v is normalized)?

Question 3

In Principal Component Analysis, if a unit vector u is constrained to be orthogonal to the first principal component, what is the maximum possible variance of the data projected onto u ?

Question 4

Let C be the covariance matrix of a dataset, and let w be a unit vector. Which expression represents the variance of the data when projected onto the direction w ?

Question 5

A centered dataset in R^{d} has a covariance matrix with eigenvalues given by λ_{i} = 80 \cdot (1/2)^{i} for i = 1, 2, \dots, d . What is the maximum possible variance of the data when projected onto any single unit direction?

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Singular Value Decomposition and Principal Component Analysis Practice Questions for GATE DA: 15+ 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 (5 Problems)

Step-by-Step Solution

Step-by-Step Solution

Step-by-Step Solution

Step-by-Step Solution

Step-by-Step Solution

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