Eigenvalues of a transposition: a) Consider the transformation T in the space M2½2 of 2 × 2 matrices, T(A) = AT. Find all its eigenvalues and eigenvectors. Is it possible to diagonalize this transformation? Hint: While it is possible to write a matrix of this linear transformation in some basis, compute characteristic polynomial, and so on, it is easier to find eigenvalues and eigenvectors directly from the definition. b) Can you do the same problem but in the space of n x n matrices?

Eigenvalues of a transposition: a) Consider the transformation T in the space M2½2 of 2 × 2 matrices, T(A) = AT. Find all its eigenvalues and eigenvectors. Is it possible to diagonalize this transformation? Hint: While it is possible to write a matrix of this linear transformation in some basis, compute characteristic polynomial, and so on, it is easier to find eigenvalues and eigenvectors directly from the definition. b) Can you do the same problem but in the space of n x n matrices?

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Question

Eigenvalues of a transposition:
a) Consider the transformation T in the space M2x2 of 2×2 matrices, T(A) =
AT. Find all its eigenvalues and eigenvectors. Is it possible to diagonalize
this transformation? Hint: While it is possible to write a matrix of this
linear transformation in some basis, compute characteristic polynomial,
and so on, it is easier to find eigenvalues and eigenvectors directly from the
definition.
b) Can you do the same problem but in the space of n x n matrices?

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