# Linear Algebra Question

Anonymous

Question description

Of course the identity matrix I ∈ Matn(F) has the basis of eigenvectors En, the standard basis of column vectors, each with eigenvalue 1. We in- troduced the elementary matrices as being “close to the identity.” That is reflected in the fact that most elements of the basis E remain eigenvectors for each elementary matrix, again with eigenvalue 1.

(a)  For Si(r), with r ̸= 1, find n − 1 elements of E that are eigenvectors for the eigenvalue 1. Find the remaining eigenvalue and associated eigen- vector.

(b)  For Xi,j find n − 2 elements of E that are eigenvectors for the eigenvalue 1. Find the remaining two eigenvalues and associated eigenvectors.

(c)  For Ri,j (a), with a ̸= 0, find n − 1 elements of E that are eigenvectors for the eigenvalue 1. Prove that this matrix cannot be diagonalized.

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