Consider a nonsymmetric real matrix A which has the Schur decomposition A = QRAH where Q...

 

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Consider a nonsymmetric real matrix A which has the Schur decomposition A = QRAH where Q is unitary, and R is upper trian- gular. (a) Show that q₁ which is the first column of Q is an eigenvector of A. Are the other columns eigenvectors? (b) Find a Schur decomposition of the matrix A₁ = A -oggf. (c) What are the eigenvalues of A₁? (d) Assume that the eigenvalues of A are such that A1 A2 A3| > |A4|>... |n| where for simplicity it is assumed that the eigenvalues are all real. Suppose you use the power method and have computed the eigenpair A₁ and u₁. Please propose a method for computing the next eigenvalue A2 by the power method applied to a certain matrix. Consider a nonsymmetric real matrix A which has the Schur decomposition A = QRAH where Q is unitary, and R is upper trian- gular. (a) Show that q₁ which is the first column of Q is an eigenvector of A. Are the other columns eigenvectors? (b) Find a Schur decomposition of the matrix A₁ = A -oggf. (c) What are the eigenvalues of A₁? (d) Assume that the eigenvalues of A are such that A1 A2 A3| > |A4|>... |n| where for simplicity it is assumed that the eigenvalues are all real. Suppose you use the power method and have computed the eigenpair A₁ and u₁. Please propose a method for computing the next eigenvalue A2 by the power method applied to a certain matrix.

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a Since Q is unitary its columns are orthonormal and thus q1 is a unit vector From the Schur decompo... View the full answer