(a) Prove that every eigenvalue of a matrix A is also an eigenvalue of its transpose AT (b) Do they have the same eigenvectors Prove that if v is an eigenvector of A with eigenvalue and w is an eigenvector of AT with a different eigenvalue , then v and w are orthogonal vectors with respect to the dot product (c) Illustrate this result when 0 I 2 1 5 4 2 (i) A ( ( 1 2 1) (ii) A54

Question: (a) Prove that every eigenvalue of a matrix A is also an eigenvalue of its transpose AT. (b) Do they have the same eigenvectors? Prove

(a) Prove that every eigenvalue of a matrix A is also an eigenvalue of its transpose AT.
(b) Do they have the same eigenvectors? Prove that if v is an eigenvector of A with eigenvalue Î» and w is an eigenvector of AT with a different eigenvalue Î¼ ‰ Î», then v and w are orthogonal vectors with respect to the dot product.
(c) Illustrate this result when

0 -I 2 1 5 -4 2 (i) A=( ^-(-1-2-1) (ii) A54

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