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ECEN601_hw9_pfister

# ECEN601_hw9_pfister - ECEN 601 Assignment 9 Problems 1 Show...

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ECEN 601: Assignment 9 Problems: 1. Show that for a square matrix F satisfying bardbl F bardbl < 1 for a norm satisfying the submultiplicative property, bardbl ( I F ) - 1 bardbl ≤ 1 1 − bardbl F bardbl . Hint: Use the Neumann expansion. 2. Show that if bardbl · bardbl is a norm satisfying the submultiplicative property and F is a matrix with bardbl F bardbl < 1, then I F is non-singular. Hint: If I F is singular, there is a vector x such that ( I F ) x = 0 . 3. Show for a square matrix F with bardbl F bardbl < 1, where the norm satisfies the submultiplicative property, that bardbl I ( I F ) - 1 bardbl ≤ bardbl F bardbl 1 − bardbl F bardbl . Hint: Show that I ( I F ) - 1 = F ( I F ) - 1 . 4. Show that for m × m matrices A and B , bardbl AB bardbl F ≤ bardbl A bardbl 2 bardbl B bardbl F . Hint: Start by showing that bardbl AB bardbl 2 F = tr( B H A H AB ). 5. Show that if A has both a left and a right inverse, they must be the same. 6. On pseudoinverses: (a) Show that, if A has linearly independent columns, its left inverse ( A H A ) - 1 A H is its pseudoinverse. (b) Show that, if A has linearly independent rows, its right inverse A H ( A H A ) - 1 is its pseu- doinverse. Optional Problems:

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ECEN601_hw9_pfister - ECEN 601 Assignment 9 Problems 1 Show...

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