(a) Find all eigenvalues and eigenvectors of the matrices A and B and normalize the eigenvectors: A = egin{pmatrix} 3 & 2 & 2 \ 1 & 4 & 1 \ -2 & -4 & -1 end{pmatrix} B = egin{pmatrix} 3 & -1 & 1 \ -1 & 3 & -1 \ 1 & -1 & 5 end{pmatrix} (b) Verify that the eigenvectors of B are orthogonal while those of A are not. (c) Produce a unitary matrix U such that UBU^{-1} is diagonal. (d) Produce a matrix P and its inverse P^{-1} such that PAP^{-1} is diagonal.
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The characteristic equation of \( A \) is given by: \[ \det(A - \lambda I) = 0 \] For \( A = \begin{pmatrix} 3 & 2 & 2 \\ 1 & 4 & 1 \\ -2 & -4 & -1 \end{pmatrix} \), we have: \[ \det \begin{pmatrix} 3 - \lambda & 2 & 2 \\ 1 & 4 - \lambda & 1 \\ -2 & -4 & -1 - Show more…
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