Linear Algebra : Eigenvalues as Optimization

Study concepts, example questions & explanations for Linear Algebra

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Example Questions

Example Question #1 : Eigenvalues As Optimization

True or False, the Constrained Extremum Theorem only applies to skew-symmetric matrices. 

Possible Answers:

True

False

Correct answer:

False

Explanation:

It only applies to symmetric matrices, not skew-symmetric ones. The Constrained Extremum Theorem concerns the maximum and minimum values of the quadratic form  when .

Example Question #2 : Eigenvalues As Optimization

The maximum value of a quadratic form  ( is an  symmetric matrix, ) corresponds to which eigenvalue of ?

Possible Answers:

The second largest eigenvalue

The eigenvalue with the greatest multiplicity

None of the other answers

The largest eigenvalue

The smallest eigenvalue

Correct answer:

The largest eigenvalue

Explanation:

This is the statement of the Constrained Extremum Theorem. Likewise, the minimum value of the quadratic form corresponds to the smallest eigenvalue of .

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