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! (Happy)

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- Feb 16th 2009, 10:05 AMbobbyProve that a matrix is non-singular
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! (Happy) - Feb 16th 2009, 10:52 AMThePerfectHacker
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.

- Feb 16th 2009, 01:06 PMbobby
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!