Finding a square root of a upper triangular matrix is not hard. Assuming that the entries on the diagonal of are real and positive, define the matrix as follows. For let Also define Let

See that Of course, this will also work for matrices with complex entries if you choose a square root for each and if the denominators in and are non-zero. Since every square matrix with complex entries is similar to an upper triangular matrix, you can use the above to find a square root of any element of

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