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Analysis of Max-Min Eigenvalue of Constrained Linear Combinations of Symmetric Matrices

This paper studies the problem whether the smallest eigenvalue of constrained linear combinations of symmetric matrices can reach a desirable value, which actually extends the mathematical problem of finding a positive definite linear combination of symmetric matrices(PDLC), and provides a universal framework to maximize the minimal eigenvalue of linear combined symmetric matrices. For solving this problem, we cast an equivalent optimization task, and propose one general algorithm framework that is proved to be globally optimal and convergent. Both theoretical analysis and experiments under a typical constraint verify our algorithm's validity and efficiency.