Thank you Let A = [1 3 3 1 0 0 0 0]. Compute the Singular Value Decomposition of

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Thank you

Let A = [1 3 3 1 0 0 0 0]. Compute the Singular Value Decomposition of A. Let A be an m times n with rank (A) = r, and singular value decomposition A = U sigma V^T. a. Prove nu_1, nu_2, ...., nu_r form an orthonormal basis for Col(A^T). b. Prove u_r+1, u_r+2, ....u_m form an orthonormal basis for Nul(A^T). Write the quadratic equation, x^2 + xy + y^2 - 6 = 0, in the form nu^T A nu = b, and use this form to find a suitable change of coordinates to plot it. Plot the equation. Prove the Spectral Theorem for a general 3 times 3 symmetric matrix, as was done in class. Is the matrix A = [1 5 0 8.1 9 1 3 0 -2 7 0 0 0 Squareroot 2 7 8 -2 2.9 3 2 9 7 7 2/3 1] positive definite? Justify your answer. Find Cholesky factorization for the positive definite matrix A = [4 2 -2 2 10 2 -2 2 5].

Explanation / Answer

A positive definite matrix is a symmetric matrix with all positive eigenvalues.

This matrix A is not symmetric in the first place so it is not a positive definite matrix.

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