The rank of A is equal to r if and only if there exists an invertible m × m matrix X and an invertible n × n matrix Y these that
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Thinking of matrices as tensors, the tensor rank generalizes to arbitrary tensors; for tensors of purchase higher than 2 (matrices are order two tensors), rank may be very difficult to compute, as opposed to for matrices.
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e., accurate football predictions the column rank of the) have to be at the very least as large as r. This proves that row rank of A is no much larger than the column rank of A. Now use this consequence to the transpose of A to have the reverse inequality and conclude as from the previous evidence. Alternative definitions[edit]
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