3. Definition. An eigenvalue of a matrix A element of Mn(R) is a scalar c such that given a linear transformation f: XY where X and Y are vector spaces over a field F with X=Y, f(x)=cx, x element of X; or alternatively, Ax=cx, and x is called the eigenvector associated with the eigenvalue c. Prove the ff:
i.c=0 is an eigenvalue of A iff A is singular.
ii.If c is an eigenvalue of A, then c^-1 is an eigenvalue of A^-1. Hence, all eigenvalues of A^-1 are the reciprocals of the eigenvalues of A.
iii.If c is an eigenvalue of A, then c is also an eigenvalue of Atranspose.
9. Given A element of Mn(R)
i.Show that det(A) is the product of all the roots of the characteristics polynomial of A.
ii.Show that tr(A) is the sum of the eigenvalues of A.
2006-12-21
18:37:10
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Mathematics