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I have read in many research papers related with iteration methods to find the generalized inverses. Where to show efficiency of the methods randomly generated matrices of higher order have been used sometime matrices with elements randomly taken from certain interval are used. I want to know why we take randomly generated matrices of higher order When simple matrices are available in literature?

I want to know importance and significance of randomly generated matrices. Please help me to understand this. I would be very much thankful.

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$\mathbf{I}$ is just an extreme example if the matrix is not randomly generated. Also, with lower order matrices, all the methods may probably work well, but you cannot extrapolate to tell the situation in high order cases. –  chaohuang Aug 29 '12 at 15:55
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There could be some other ways, but creating random matrices may be the simplest, thus most common used approach. And it's sufficient to tell the performance of a method, unless you're so unlucky. –  chaohuang Aug 29 '12 at 16:21
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A random matrix models a real-life input. You're writing a method and expecting that a random user calls it with some input. You don't know if the input is just a special case (e.g. zero matrix, identity matrix, triangular matrix, full rank matrix etc) or more general. So you assume it's just random. I think there is a more formal statement about the expected behaviour of any algorithm over a set of input drawn from a random distribution. –  user2468 Aug 31 '12 at 13:37
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He is a much worded version of my comment above. To establish a statement about correctness/efficiency of an algorithm, we need to test with all possible inputs which is impractical/impossible. But we can test with a sample of the input space. Here comes the catch phrase: with random samples we can establish "statistically meaningful estimates" of the program bahvior over the whole input space. (quote from this PPT) –  user2468 Aug 31 '12 at 13:57
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For you other question. Frankly, I'm not sure. The uniformly distributed radnom elements are typically generated from a finite interval. In computational finance & machine learning, for example, I've read that they encourage to normalize all inputs to the interval $[-1, +1]$. There was a good reason for this, but I forgot it :( The mean is zero or something?. See for example this question. –  user2468 Aug 31 '12 at 13:57
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