GATE CE 2023 Set 1 — Question 24
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Engineering Mathematics → Linear Algebra → Eigenvalues & Eigenvectors
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Question
If is an arbitrary real matrix, then which of the following matrices will have non-negative eigenvalues?
Correct answer
(B) MM^T; (C) M^TM
Solution
A real matrix has non-negative eigenvalues if it is positive semi-definite (PSD).
For any real matrix , the product of the matrix and its transpose ( and ) results in a symmetric positive semi-definite matrix.Proof for :
For any non-zero vector , the quadratic form is:
.
Since the quadratic form is always non-negative, is positive semi-definite, and all its eigenvalues are real and non-negative. The same logic applies to . and do not necessarily have non-negative eigenvalues unless itself has specific properties (e.g., if is symmetric). For an arbitrary real matrix, can have negative or complex eigenvalues.
For any real matrix , the product of the matrix and its transpose ( and ) results in a symmetric positive semi-definite matrix.Proof for :
For any non-zero vector , the quadratic form is:
.
Since the quadratic form is always non-negative, is positive semi-definite, and all its eigenvalues are real and non-negative. The same logic applies to . and do not necessarily have non-negative eigenvalues unless itself has specific properties (e.g., if is symmetric). For an arbitrary real matrix, can have negative or complex eigenvalues.
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