Optimization Principles for Singular Values: Let A be any nonzero m ×n matrix. Prove that (a) σ1

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Optimization Principles for Singular Values: Let A be any...

Optimization Principles for Singular Values: Let $A$ be any nonzero $m \times n$ matrix. Prove that (a) $\sigma_1=\max \{\|A \mathbf{u}\| \mid\|\mathbf{u}\|=1\}$. (b) Is the minimum the smallest singular value? (c) Can you design optimization principles for the intermediate singular values?

Key Concepts

Singular Value Decomposition (SVD)

SVD is a factorization of a matrix into the form A = U?V*, where U and V are unitary (or orthogonal in the real case) matrices and ? is a diagonal matrix containing the singular values. These singular values, which are nonnegative, play a central role in understanding the action of the matrix, particularly in quantifying its “stretching” effect along certain directions. They form the foundation for many optimization principles involving matrices.

Operator Norm

The operator norm (or induced norm) of a matrix is defined as the maximum norm of the image of any unit vector under the matrix transformation. In the context of singular values, the largest singular value corresponds exactly to this operator norm, providing a direct optimization interpretation: it is given by the maximization problem of ||A u|| over all unit vectors u.

Optimization Characterization of Singular Values

This concept involves expressing singular values as the result of optimization problems. For example, the largest singular value is obtained by maximizing ||A u|| over all unit vectors u. Similarly, through minimax principles such as the Courant-Fischer theorem, the intermediate singular values can be characterized using nested optimization problems over subspaces, where one seeks to maximize the minimum gain over all vectors in a constrained subspace.

Minimax (Courant-Fischer) Principle

The minimax principle generalizes the idea of optimizing the largest or smallest singular values to the intermediate ones. By considering optimizations over subspaces of appropriate dimensions, one can characterize each singular value as the maximum of a minimum (or the minimum of a maximum) of ||A u|| over unit vectors outside of certain subspaces. This approach reveals the structure of the singular values in a way analogous to eigenvalues of symmetric matrices.

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