I am reading a paper and I saw:

AFFHAHAFF^{H}A^{H} is singular, where A∈CN×MA \in \mathbb{C}^{N\times M}, F∈M×1F \in \mathbb{}^{M\times1} and N>MN > M

So I wonder why is that so. My naive guessing is: Since FFHFF^{H} is in CM×M\mathbb{C}^{M\times M}, AFFHAHAFF^{H}A^{H} will have an M×MM\times M non-singular submatrix, but the rest of them (N−MN-M) will be a linear combination of MM elements.

However, I am not sure whether my guess is correct or not. Even if my guess is correct, it is not somewhat strictly defined. Could anyone show me a right way?

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2 Answers

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You may argue in terms of dimensions and rank. The dimension of the image of a product of matrices is never greater than the minimal rank of each factor. In the present case you seem to say (but there is a misprint) that FF is of rank 1. So the full product is at most of rank one. In particular, there is at most one non-zero eigenvalue.

Well then, the statement above is always true regardless of the condition N>MN>M?

– Jeon

2 days ago

Yes, if N>1N>1 (and if I understood correctly the def of FF).

– H. H. Rugh

2 days ago

Note that XXHXX^H and XHXX^HX have the same rank. Then AFFHAHAFF^HA^H has the same rank as

FHAHAF≤‖AHA‖FHF,

F^HA^HAF\leq\|A^HA\|\,F^HF,

and FHFF^HF is rank-one.