Singular Values and Gram Matrices Short Notes for GATE DA: Concepts, Formulas, Worked Examples & Practice

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 (2 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.

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Singular Values and Gram Matrices Short Notes for GATE DA: Concepts, Formulas, Worked Examples & Practice

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

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) ?

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Singular Values and Gram Matrices Short Notes for GATE DA: Concepts, Formulas, Worked Examples & Practice

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 (2 Problems)

Step-by-Step Solution

Step-by-Step Solution

More short notes in this unit

A question from this chapter

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Positive Definiteness (Symmetric $A$ )