Singular Values and Gram Matrices Practice Questions for GATE DA: 36+ Solved Questions with Step-by-Step Solutions

Solve 36+ Singular Values and Gram Matrices practice questions for GATE DA with answers and detailed solutions. Free sample questions below.

Chapter Roadmap: Singular Values and Gram Matrices

Chapter Roadmap

Singular Values and Gram Matrices

1. Gram Matrices and Positive Definiteness

Inner products, geometry of vector sets, and conditions for positive definiteness.

2. Singular Values and Spectral Properties

SVD, pseudo-inverses, and spectral properties of special matrices.

What You Will Master

Construct and analyze Gram matrices to determine vector independence.
Rigorously test matrices for positive definiteness using Sylvester's criterion.
Decompose any matrix into its singular values and understand its geometric action.

Intuition: Measuring Geometry with Gram Matrices

Intuition: Measuring Geometry

Imagine you have a set of vectors in space. How do you capture all their lengths and the angles between them in a single, compact object?

The Gram matrix does exactly this.

It is a matrix composed entirely of inner products (dot products) between the vectors. Instead of looking at the vectors individually, the Gram matrix encodes the entire geometric relationship of the set. It tells you how much each vector "overlaps" with every other vector.

Singular Values and Gram Matrices: Solved Questions with Step-by-Step Explanations (5 Problems)

Question 1 · Linear Algebra MCQ

Which of the following properties is a necessary condition for any matrix to be a valid Gram matrix of a set of real vectors?

A.
It must be skew-symmetric
B.
It must have all negative eigenvalues
C.
It must be an orthogonal matrix
D.
It must be positive semi-definite

Correct Answer:

Step-by-Step Solution

Key idea: This tests the fundamental properties of Gram matrices.

Step 1: Let $X$ be the matrix whose columns are the given real vectors. The Gram matrix is $G = X^{T} X$ .

Step 2: For any real vector $v$ , the quadratic form is $v^{T} G v = v^{T} X^{T} X v = (X v)^{T} (X v) = ∥ X v ∥^{2}$ .

Step 3: Since the squared norm $∥ X v ∥^{2}$ is always greater than or equal to zero, $v^{T} G v \geq 0$ for all $v$ .

Step 4: This is the exact definition of a positive semi-definite matrix.

Answer: Option D is correct.

Question 2 · Linear Algebra MCQ

Three linearly independent vectors in $R^{3}$ are used to form a Gram matrix $G$ . What is the sign of $det (G)$ ?

A.
Strictly negative
B.
Zero
C.
Strictly positive
D.
Cannot be determined

Correct Answer:

Step-by-Step Solution

Key idea: This is a direct-application question linking linear independence to the Gram determinant.

Step 1: Let the three linearly independent vectors be columns of a $3 \times 3$ matrix $X$ .

Step 2: Since the vectors are independent, $X$ is full rank, so $det (X) \neq = 0$ .

Step 3: The Gram matrix is $G = X^{T} X$ .

Step 4: Using the multiplicative property of determinants: $det (G) = det (X^{T} X) = det (X^{T}) det (X) = (det (X))^{2}$ .

Step 5: Since $det (X) \neq = 0$ , we have $(det (X))^{2} > 0$ .

Step 6: Therefore, $det (G)$ is strictly positive.

Answer: Option C is correct.

Question 3 · Linear Algebra MCQ

The singular values of a real matrix $A$ are defined as the square roots of the eigenvalues of which of the following matrices?

A.
$A^{T} A$
B.
$A + A^{T}$
C.
$A - A^{T}$
D.
$A^{2}$

Correct Answer:

Step-by-Step Solution

Key idea: This is a definition-recall question on the relationship between singular values and eigenvalues.

Step 1: Recall the definition. The singular values $σ_{i}$ of a real matrix $A$ are defined as $σ_{i} = λ_{i} (A^{T} A)$ , where $λ_{i} (A^{T} A)$ are the eigenvalues of $A^{T} A$ .

Step 2: Note that $A^{T} A$ is always symmetric and positive semi-definite, so its eigenvalues are always real and non-negative. This guarantees that the square roots are well-defined real numbers.

Step 3: Check the other options. $A + A^{T}$ is symmetric but its eigenvalues can be negative. $A - A^{T}$ is skew-symmetric with purely imaginary eigenvalues. $A^{2}$ may not even be defined for non-square matrices.

Answer: Option A is correct.

Question 4 · Linear Algebra MCQ

If the eigenvalues of the matrix $A^{T} A$ are $16$ and $9$ , what are the singular values of $A$ ?

A.
$16$ and $9$
B.
$4$ and $3$
C.
$256$ and $81$
D.
$8$ and $6$

Correct Answer:

Step-by-Step Solution

Key idea: This is a direct-formula question connecting the singular values of $A$ to the eigenvalues of $A^{T} A$ .

Step 1: Recall the fundamental relationship: the singular values $σ_{i}$ of a matrix $A$ are the square roots of the eigenvalues $λ_{i}$ of $A^{T} A$ .

Step 2: The formula is $σ_{i} = λ_{i} (A^{T} A)$ .

Step 3: The given eigenvalues of $A^{T} A$ are $16$ and $9$ .

Step 4: Calculate the singular values: $σ_{1} = 16 = 4$ and $σ_{2} = 9 = 3$ .

Step 5: The singular values are $4$ and $3$ .

Answer: Option B is correct.

Question 5 · Linear Algebra MCQ

If $A$ is a real symmetric matrix, what is the geometric relationship between any two eigenvectors of $A$ that correspond to distinct eigenvalues?

A.
They are parallel
B.
They are orthogonal
C.
They are linearly dependent
D.
They have the same magnitude

Correct Answer:

Step-by-Step Solution

Key idea: This is a direct-recall question on the spectral properties of real symmetric matrices.

Step 1: Recall the Spectral Theorem for real symmetric matrices. It states that a real symmetric matrix has real eigenvalues and its eigenvectors corresponding to distinct eigenvalues are orthogonal.

Step 2: Let $λ_{1} \neq = λ_{2}$ be distinct eigenvalues with eigenvectors $v_{1}, v_{2}$ .

Step 3: The property $λ_{1} (v_{1}^{T} v_{2}) = (A v_{1})^{T} v_{2} = v_{1}^{T} A^{T} v_{2} = v_{1}^{T} A v_{2} = v_{1}^{T} (λ_{2} v_{2}) = λ_{2} (v_{1}^{T} v_{2})$ .

Step 4: Since $λ_{1} \neq = λ_{2}$ , we must have $v_{1}^{T} v_{2} = 0$ .

Step 5: Therefore, the eigenvectors are orthogonal.

Answer: Option B is correct.

A question from this chapter

Question 1

Which of the following properties is a necessary condition for any matrix to be a valid Gram matrix of a set of real vectors?

Question 2

Three linearly independent vectors in R^{3} are used to form a Gram matrix G . What is the sign of det (G) ?

Question 3

The singular values of a real matrix A are defined as the square roots of the eigenvalues of which of the following matrices?

Question 4

If the eigenvalues of the matrix A^{T} A are 16 and 9, what are the singular values of A ?

Question 5

If A is a real symmetric matrix, what is the geometric relationship between any two eigenvectors of A that correspond to distinct eigenvalues?

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Singular Values and Gram Matrices Practice Questions for GATE DA: 36+ Solved Questions with Step-by-Step Solutions

Chapter Roadmap: Singular Values and Gram Matrices

Chapter Roadmap

1. Gram Matrices and Positive Definiteness

2. Singular Values and Spectral Properties

Intuition: Measuring Geometry with Gram Matrices

Intuition: Measuring Geometry

Singular Values and Gram Matrices: 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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