Hey, i am really stuck with this question and any help would be greatly appreciated!
If D is a diagonal matrix with positive diagonal entries and L is a lower triangular matrix, I have to show that D-L is non-singular.
Thanks in advance!
Hey, i am really stuck with this question and any help would be greatly appreciated!
If D is a diagonal matrix with positive diagonal entries and L is a lower triangular matrix, I have to show that D-L is non-singular.
Thanks in advance!
This is not true unless $\displaystyle L$ is a strictly lower triangular matrix. Otherwise take $\displaystyle D=L$ then then $\displaystyle D-L$ is a zero matrix which is definitely not invertible. However, if $\displaystyle L$ is a stricly lower triangular matrix then $\displaystyle D-L$ is a lower triangular matrix with non-zero diagnol entries. Now use the fact that the determinant of a triangular matrix is the product of its entires diagnol entries. Therefore, the determinant is non-zero.
thanks for your answer!
I missed out a part of the question which may be crucial as L is just lower triangular.
Suppose A is positive definite. Note that we can write A = D - L - L* with D and L defined in my first post and L* is the conjugate transpose of L i think. Show that D - L is non-singular.
Thanks very much!