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Let $\pmb{H}$ be a matrix where $\pmb{H}=[\pmb{H}_{1}, \pmb{H}_{2}, ...,\pmb{H}_{N}]$ where $H_{N} \in \mathbb{C}^{K\times R}$ and $K > R$, and all entries of $\pmb{H}_{N}$ are randomly generated between ($0-1$). How to design $\pmb{H}$ or how to make $\pmb{H}$ as $\pmb{H}_{i}^{H}\pmb{H}_{j}=0, i \neq j$ which means satisfy orthogonality.

One of the solutions is Gram-Schmidt orthogonalization but the complexity is high $KN^{2}$ and in my case $K, N$ are very big. Could you Please help me with a solution that satisfies the earlier condition and lower complexity?

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    1) Why the indices 1 and 2 ? 2) IMHO you can't lower $O(N^2)$ complexity, unless maybe $K >> R$. Besides, what is $N$ ? $N=max(K,R)$ ? – 2017-01-11
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    Yes $K >> R$, Generally, $\pmb{H}_{i}^{H}\pmb{H}_{j}=0$ where $i \neq j$ deos not mean that only $\pmb{H}_{1}^{H}\pmb{H}_{2}=0$ – 2017-01-11
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    Do you only need orthogonality, not normed columns (column norms=1) ? – 2017-01-11
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    only need orthogonality, does not matter if the norm 1 or not – 2017-01-11
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    Also, $N$ is only the number of $\pmb{H}_{N}$, like for example if you say I have $6$ balls then $N=6$ but every ball has a dimension $K \times R$ and K >> R – 2017-01-11
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    A last thing: it is not advisable to use the name "orthogonal matrix" for a matrix that has only columns' orthogonality: see (https://en.wikipedia.org/wiki/Orthogonal_matrix) – 2017-01-11
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    Really, appreciate your helping but I didn't get it very well your answer if you don't mind please to explain more or recommend me a paper or book that have the prove or more examples? – 2017-01-11
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    Are you accustomed to Matlab ? I could give you a more precise answer in some hours. – 2017-01-11
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    Yes, I do use Matlab. I will wait for your precise answer thank you in advance – 2017-01-11
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    I have suppressed my remark about product KH. It is not interesting. What I would advise you to try: construct once for all a big (portrait format) matrix with orthogonal columns, for example by using "orth" in Matlab. Then scramble its lines, by a random permutation (use Matlab's "randperm"): this will preserve orthogonality. Of course, there will be a small correlation between the new and the old matrix. As I don't know the use you want to do of these matrices, will it be harmful ? – 2017-01-11
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    Maybe I understood you wrong but The problem is I can not construct $\pmb{H}$ as an initial step because it comes as an output result from an algorithm but once I receive it then I can play with it. Also, I can't reconstruct it again. As I mentioned earlier Gram-Schmidt allows me to take every $\pmb{H}_{i}$ independently to make them orthogonal. Sorry if my answer out of the scope – 2017-01-11
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    You mean that you want to preserve the image space (I.e., column space) but with an orthogonal basis for it ? – 2017-01-11
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    So $H$ consists of $N$ block matrices $H_i$, each having $K\times R$ complex entries? So $H$ is a $K\times(N\,R)$ matrix? It seems unusual to me to call those blocks 'columns'. – 2017-01-12
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    Hi JeanMarie, Yes exactly that what I am trying to do. – 2017-01-12
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    +mvm, It seems like that but if you use Gram-Schmidt it is possible to make $\pmb{H}_{i}^{H}\pmb{H}_{j}=0$. So, this is the reason which is encouraged me try to find another solution – 2017-01-12

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