Wing-Tat Chiu wrote:
>I am trying to find the matrix notation for the elementwise inverse of
>an upper triangular matrix. Do you happen to be aware of such notation?
>
>Example:
>
>A = [
>4 9 16;
>0 4 16;
>0 0 9]
>
>Desired outcome = [
> 1/9 1/16;
>0 1/4 1/16;
>0 0 1/9]
>
>OR
>
>Desired outcome = [
>0.25 0.11 0.06;
>0 0.25 0.06;
>0 0 0.11]
For the sake of precision in this example, it's better to use rational
numbers for the elements; in any case, the inverse matrix of A is the
matrix adjoint of A [i.e. the transpose of the matrix of its cofactors]
divided by the determinant of A. With an upper triangular matrix, its
matrix of cofactors is a lower triangular matrix, so its adjoint is also
an upper triangular matrix; and its determinant is identical to its
trace [i.e. the product of its diagonal elements], making the
computations a bit simpler than in the general case.
Note that your results for the off-diagonal elements of the "desired
outcome" matrix are not correct, and that two of them should have a
negative sign.
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