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Math Help - Prove that a matrix is non-singular

  1. #1
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    Prove 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!
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  2. #2
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    Quote Originally Posted by bobby View Post
    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 L is a strictly lower triangular matrix. Otherwise take D=L then then D-L is a zero matrix which is definitely not invertible. However, if L is a stricly lower triangular matrix then 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.
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  3. #3
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    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!
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