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Singular Values and Gram Matrices Short Notes for GATE DA

Singular Values and Gram Matrices short notes for GATE DA: 2 study cards covering concepts, formulas, shortcuts and exam traps, plus solved practice questions

singular values and gram matrices short notes

Gram Matrices and Positive Definiteness Cheat Sheet

Cheat Sheet Gram Matrices Definition: G_{ij} = x_{i}^{T} x_{j} . Matrix form: G = X^{T} X . Core Properties: Always symmetric, always Positive Semi-Definite. Linear Independence: G is strictly Positive Definite iff vectors are independent. Gram Determinant: det (G) equals the volume of the spanned parallelepiped. Positive Definiteness (Symmetric A) Definition: x^{T} A x > 0 for all x \neq = 0 . Eigenvalues: All λ_{i} > 0 . Sylvester's: All leading principal minors > 0 . Cholesky: A = L L^{T} exists. Gram Form: A = X^{T} X for a full-rank matrix X .

SVD and Spectral Properties Cheat Sheet

SVD and Spectral Properties Cheat Sheet Singular Value Decomposition (SVD) Formula: A = U Σ V^{T} (exists for all matrices). Singular Values: σ_{i} = λ_{i} (A^{T} A) . Rank Connection: Rank (A) = count of non-zero σ_{i} . Geometry: V^{T} rotates, Σ stretches, U rotates. Spectral Properties of Special Matrices Symmetric (A = A^{T}): Real eigenvalues, orthogonal eigenvectors (Spectral Theorem). Skew-Symmetric (A = - A^{T}): Purely imaginary or zero eigenvalues, zero diagonal. Orthogonal (Q^{T} Q = I): Eigenvalues have magnitude 1 (∣ λ ∣ = 1). Normal (A^{T} A = A A^{T}): Unitarily/orthogonally diagonalizable (includes symmetric, skew-symmetric, orthogonal).

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Question 1

Level 1: Warm-up

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

Level 1: Warm-up

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

Question 3

Level 1: Warm-up

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

Level 1: Warm-up

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

Question 5

Level 1: Warm-up

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

Question 6

Level 1: Warm-up

For a general real matrix A that is NOT symmetric, which of the following relationships between its singular values σ_{i} and its eigenvalues λ_{i} is always true?

Question 7

Level 1: Warm-up

For a real symmetric matrix A, which of the following conditions is equivalent to A being positive definite?

Question 8

Level 1: Warm-up

A real symmetric matrix A satisfies x^{T} A x > 0 for every non-zero vector x \in R^{n} . By definition, what is such a matrix called?

Question 9

Level 1: Warm-up

If a real symmetric matrix A has eigenvalues λ_{1} = 2, λ_{2} = - 3, λ_{3} = 5, then A is:

Question 10

Level 1: Warm-up

A 5 \times 4 matrix A has exactly two non-zero singular values. What is the rank of A ?

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Singular Values and Gram Matrices Short Notes for GATE DA

Singular Values and Gram Matrices short notes for GATE DA: 2 study cards covering concepts, formulas, shortcuts and exam traps, plus solved practice questions.

Gram Matrices and Positive Definiteness Cheat Sheet

Cheat Sheet

Gram Matrices

Definition: $G_{ij} = x_{i}^{T} x_{j}$ . Matrix form: $G = X^{T} X$ .
Core Properties: Always symmetric, always Positive Semi-Definite.
Linear Independence: $G$ is strictly Positive Definite iff vectors are independent.
Gram Determinant: $det (G)$ equals the volume of the spanned parallelepiped.

Positive Definiteness (Symmetric $A$ )

Definition: $x^{T} A x > 0$ for all $x \neq = 0$ .
Eigenvalues: All $λ_{i} > 0$ .
Sylvester's: All leading principal minors $> 0$ .
Cholesky: $A = L L^{T}$ exists.
Gram Form: $A = X^{T} X$ for a full-rank matrix $X$ .

SVD and Spectral Properties Cheat Sheet

Singular Value Decomposition (SVD)

Formula: $A = U Σ V^{T}$ (exists for all matrices).
Singular Values: $σ_{i} = λ_{i} (A^{T} A)$ .
Rank Connection: $Rank (A) = count of non-zero σ_{i}$ .
Geometry: $V^{T}$ rotates, $Σ$ stretches, $U$ rotates.

Spectral Properties of Special Matrices

Symmetric ( $A = A^{T}$ ): Real eigenvalues, orthogonal eigenvectors (Spectral Theorem).
Skew-Symmetric ( $A = - A^{T}$ ): Purely imaginary or zero eigenvalues, zero diagonal.
Orthogonal ( $Q^{T} Q = I$ ): Eigenvalues have magnitude $1$ ( $∣ λ ∣ = 1$ ).
Normal ( $A^{T} A = A A^{T}$ ): Unitarily/orthogonally diagonalizable (includes symmetric, skew-symmetric, orthogonal).

Singular Values and Gram Matrices: Solved Questions with Step-by-Step Explanations (10 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.

Question 6 · Linear Algebra MCQ

For a general real matrix $A$ that is NOT symmetric, which of the following relationships between its singular values $σ_{i}$ and its eigenvalues $λ_{i}$ is always true?

A.
$σ_{i} = ∣ λ_{i} (A) ∣$ for all $i$
B.
$σ_{i} = λ_{i} (A^{T} A)$ for all $i$
C.
$σ_{i} = λ_{i} (A)$ for all $i$
D.
$σ_{i} = λ_{i} (A^{T} A)$ for all $i$

Correct Answer:

Step-by-Step Solution

Key idea: This is a comparison question testing the definitional link between singular values and eigenvalues, and whether the student knows which formula holds universally versus only for special matrices.

Step 1: Recall the definition of singular values. By definition, the singular values of any matrix $A$ are $σ_{i} = λ_{i} (A^{T} A)$ , where $λ_{i} (A^{T} A)$ are the eigenvalues of the symmetric positive semi-definite matrix $A^{T} A$ . This holds for every matrix, symmetric or not.

Step 2: Check Option A. The relation $σ_{i} = ∣ λ_{i} (A) ∣$ holds only for normal matrices (symmetric, orthogonal, skew-symmetric). For a general non-symmetric matrix, this is false.

Step 3: Check Option C. Singular values are always non-negative, while eigenvalues of a general matrix can be negative or complex. So $σ_{i} = λ_{i} (A)$ is false in general.

Step 4: Check Option D. This omits the square root. The eigenvalues of $A^{T} A$ are the squares of the singular values, not the singular values themselves.

Answer: Option B is correct.

Question 7 · Linear Algebra MCQ

For a real symmetric matrix $A$ , which of the following conditions is equivalent to $A$ being positive definite?

A.
All eigenvalues of $A$ are strictly positive
B.
All eigenvalues of $A$ are non-negative
C.
The determinant of $A$ is zero
D.
The trace of $A$ is negative

Correct Answer:

Step-by-Step Solution

Key idea: This is a core property question linking positive definiteness to eigenvalues.

Step 1: By definition, a symmetric matrix $A$ is positive definite if $x^{T} A x > 0$ for all non-zero vectors $x$ .

Step 2: For an eigenvector $v$ with eigenvalue $λ$ , we have $v^{T} A v = v^{T} (λ v) = λ ∥ v ∥^{2}$ .

Step 3: Since $v \neq = 0$ , $∥ v ∥^{2} > 0$ . For $v^{T} A v > 0$ to hold, we must have $λ > 0$ .

Step 4: This must be true for all eigenvectors, so all eigenvalues must be strictly positive.

Answer: Option A is correct.

Question 8 · Linear Algebra MCQ

A real symmetric matrix $A$ satisfies $x^{T} A x > 0$ for every non-zero vector $x \in R^{n}$ . By definition, what is such a matrix called?

A.
Positive semi-definite
B.
Positive definite
C.
Orthogonal
D.
Skew-symmetric

Correct Answer:

Step-by-Step Solution

Key idea: This is a definition-recall question on positive definiteness.

Step 1: The condition $x^{T} A x > 0$ for all $x \neq = 0$ is the definition of a positive definite matrix.

Step 2: Positive semi-definite would require $x^{T} A x \geq 0$ (allowing equality for some non-zero $x$ ).

Step 3: Orthogonal matrices satisfy $A^{T} A = I$ , which is unrelated to the sign of $x^{T} A x$ .

Step 4: Skew-symmetric matrices satisfy $A^{T} = - A$ , and for these $x^{T} A x = 0$ for all $x$ .

Answer: Option B is correct.

Question 9 · Linear Algebra MCQ

If a real symmetric matrix $A$ has eigenvalues $λ_{1} = 2, λ_{2} = - 3, λ_{3} = 5$ , then $A$ is:

A.
Positive Definite
B.
Negative Definite
C.
Indefinite
D.
Positive Semi-Definite

Correct Answer:

Step-by-Step Solution

Key idea: Classifying a matrix based on the signs of its eigenvalues.

Step 1: Recall the classification rules for symmetric matrices using eigenvalues.

Step 2: Positive Definite: All $λ_{i} > 0$ .

Step 3: Negative Definite: All $λ_{i} < 0$ .

Step 4: Positive Semi-Definite: All $λ_{i} \geq 0$ (at least one 0).

Step 5: Indefinite: Some $λ_{i} > 0$ and some $λ_{j} < 0$ .

Step 6: Here, we have $λ_{1} = 2 > 0$ and $λ_{2} = - 3 < 0$ .

Step 7: Since there are both positive and negative eigenvalues, $A$ is Indefinite.

Question 10 · Linear Algebra NAT

A $5 \times 4$ matrix $A$ has exactly two non-zero singular values. What is the rank of $A$ ?

Correct Answer:

Step-by-Step Solution

Key idea: This is a direct-recall question linking the rank of a matrix to its singular values.

Step 1: Recall the fundamental relationship between rank and singular values. The rank of a matrix is exactly equal to the number of its non-zero singular values.

Step 2: The problem states that the matrix $A$ has exactly two non-zero singular values.

Step 3: Therefore, the rank of $A$ is exactly $2$ .

Step 4: The dimensions of the matrix ( $5 \times 4$ ) are extra information and do not change this fundamental property.

Answer: 2

singular values and gram matrices short notes

Gram Matrices and Positive Definiteness Cheat Sheet

Cheat Sheet

Gram Matrices

Positive Definiteness (Symmetric A)

SVD and Spectral Properties Cheat Sheet

SVD and Spectral Properties Cheat Sheet

Try a question

Login to view the complete short notes

Personalised first. High quality throughout.

Built around you, not around a syllabus PDF

Revision that hits your weak spots

Questions calibrated to the real exam

Notes written for recall, not for volume

One place for everything

Honest progress

Unlock the whole course

Singular Values and Gram Matrices Short Notes for GATE DA

Gram Matrices and Positive Definiteness Cheat Sheet

Cheat Sheet

Gram Matrices

Positive Definiteness (Symmetric A)

SVD and Spectral Properties Cheat Sheet

SVD and Spectral Properties Cheat Sheet

Singular Values and Gram Matrices: Solved Questions with Step-by-Step Explanations (10 Problems)

Step-by-Step Solution

Step-by-Step Solution

Step-by-Step Solution

Step-by-Step Solution

Step-by-Step Solution

Step-by-Step Solution

Step-by-Step Solution

Step-by-Step Solution

Step-by-Step Solution

Step-by-Step Solution

More short notes in this unit

Positive Definiteness (Symmetric $A$ )

Positive Definiteness (Symmetric $A$ )