The GATE Grind

GATE 2021 MN – Question 22

Engineering Mathematics · Linear Algebra · 1 mark · Multiple choice

Which statement is true for a real symmetric positive-definite square matrix?

  1. It is singular.
  2. All eigenvalues are greater than zero.
  3. All eigenvalues are zero.
  4. Some eigenvalues can be negative.

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Show answer and explanation

Correct answer: (B) All eigenvalues are greater than zero.

Explanation

For an eigenvector v with eigenvalue λ, positive definiteness gives vᵀAv=λvᵀv>0. Since vᵀv>0, λ>0 for every eigenvalue, B. Consequently the matrix is nonsingular; a zero or negative eigenvalue would violate positive definiteness.

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