[Solved] MA502 Homework 10

Write down detailed proofs of every statement you make

1. Let A be a real n n matrix with an eigenvalue having algebraic multiplicity n. Prove that for any t real one has

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2. Let A denote the matrix

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• Find an orthogonal matrix O such that O^T AO is diagonal
• Compute the matrix e^A.

3. Consider the vector space of polynomials with real coefficients and withinner product

Apply the Graham-Schmidt process to find an orthonormal basis, with respect to this inner product, for the subspace generated by.

4. Let A be a real n n Define . Find necessary and sufficient conditions on A for this operation to be a inner product on R³.
5. Show that the system Ax = b has no solution and find the least square solution of the problem Ax b with

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