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Exercise 20: DiagonalizationDiagonalize the matrix.How is thi... | Filo

Exercise 20: DiagonalizationDiagonalize the matrix.How is thi... | Filo

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SOLVED: Is a diagonal matrix similar only to itself?

SOLVED: Is a diagonal matrix similar only to itself?

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The nonsingular matrix P such that P − 1 A P is diagonal and al... | Filo

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SOLVED:It is pretty obvious that the trace of a diagonal matrix is the sum of its eigenvalues, and its determinant is their product (see Equation 3.78 ). It follows (from Equations 3.64 and 3.67 ) that the same holds for any diagonalizable matrix. Prove that det(𝐓)=λ1 λ2 ⋯λn+ Tr(𝐓)=λ1+λ2+⋯+λn for any matrix. (The λ's are the n solutions to the characteristic equation-in the case of multiple roots, there may be fewer linearly independent eigenvectors than there are solutions, but we still cou

SOLVED:It is pretty obvious that the trace of a diagonal matrix is the sum of its eigenvalues, and its determinant is their product (see Equation 3.78 ). It follows (from Equations 3.64 and 3.67 ) that the same holds for any diagonalizable matrix. Prove that det(𝐓)=λ1 λ2 ⋯λn+ Tr(𝐓)=λ1+λ2+⋯+λn for any matrix. (The λ's are the n solutions to the characteristic equation-in the case of multiple roots, there may be fewer linearly independent eigenvectors than there are solutions, but we still cou

(Schur's decomposition theorem) Let A be an n ×n matrix. Then there exist a unitary n ×n matrix S (that is, S^* S=In ) and an upper triangular matrix M whose diagonal elements are the eigenvalues of A, such that S^* A S=M. (a) Follow the steps of the solution to Exercise 7.46 to prove this result for n=2. (b) Then use induction to prove the general result. | Numerade

(Schur's decomposition theorem) Let A be an n ×n matrix. Then there exist a unitary n ×n matrix S (that is, S^* S=In ) and an upper triangular matrix M whose diagonal elements are the eigenvalues of A, such that S^* A S=M. (a) Follow the steps of the solution to Exercise 7.46 to prove this result for n=2. (b) Then use induction to prove the general result. | Numerade

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