Frame Diagonalization of Matrices

Pandit, Vishal Kumar and G, Ramesh (2017) Frame Diagonalization of Matrices. Masters thesis, Indian Institute of Technology.

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Abstract

This thesis introduces a concept of frame which is more than a basis in terms of linearly dependent / independent. We defines frame bound, frame operator and its properties. We discuss existence frame and when a frame is called tight. Then we discuss relation between frames bounds and eigenvalues and existence frames in C n and its properties. Also we discuss frame diagonalization and prove the ex- istence of a universal diagonalizer for each n ∈ N that simultaneously diagonalizes all matrices in M n ( C ) , and create a method of frame diagonalization that works for any matrix in M n ( C ) , uses at most [3 n/ 2] frame vectors and retains information about the eigenvalues of the matrix. At last we discuss sharp frame diagonaliza- tion. From frame diagonalization, we know that if M = Γ 1 ⊕ Γ 2 ⊕ ... ⊕ Γ k ∈ C n × n is a Jordan matrix with k nontrivial Jordan blocks Γ i , then M can be frame diag- onalized by embedding M into a diagonalizable matrix in C ( n + l ) × ( n + l with l = k . This naturally motivates a pursuit of the best possible value of l for which this is possible. Here, we use Lidskiis Theorem on eigenvalue perturbations to construct diagonalizing frames for l = k ? = max { gm M ( λ ) | λ ∈ σ ( M ) } . Moreover, we verify that k ? is sharp.

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