Let be an symmetric tridiagonal matrix with its sub- and superdiagonals nonzero. Prove that the eigenvalues of A are distinct by answering parts (a)-(e). If is an eigenvalue, show that the rank of...

Let

be an
symmetric tridiagonal matrix with its sub- and superdiagonals nonzero. Prove that the eigenvalues of A are distinct by answering parts (a)-(e).

If

is an eigenvalue, show that the rank of

is at most

Consider the upper triangular

submatrix

Show that

has rank

Show that rank
rank

Show that the null space of

is spanned by an eigenvector corresponding to

Prove that the symmetry of

implies that

must be distinct.