Eigenvalues as Optimization

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Linear Algebra › Eigenvalues as Optimization

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

False

True

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 $\mathbf{x}^TA\mathbf{x}$ when $|\mathbf{x}|=1$ .

The maximum value of a quadratic form $\mathbf{x}^TA\mathbf{x}$ ( is an $n \times n$ symmetric matrix, $|\mathbf{x}|=1$ ) corresponds to which eigenvalue of ?

The largest eigenvalue

The smallest eigenvalue

The eigenvalue with the greatest multiplicity

The second largest eigenvalue

None of the other answers

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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