Solution of Linear Equations
1. As in Example 6.8, characterize all left inverses of a matrix A IRmn .
Answer 6.1 A has a left inverse AT has a right inverse
R(In ) R(AT ) AT (AT )+ In = In rank(AT ) = r = n
(since r n) AT is onto (AT )+ is the
Eigenvalues and Eigenvectors
1. Let A Cnn have distinct eigenvalues 1 , . . . , n with corresponding
right eigenvectors x1 , . . . , xn and left eigenvectors y1 , . . . , yn , respectively. Let v Cn be an arbitrary vector. Show that v can be exp
1. Show that if a triangular matrix is normal, then it must be diagonal.
Answer 10.1 We prove this by induction on n. Let T Cnn be
normal and, without loss of generality, assume it is upper triangular.
For n = 1, the matrix T is