if A is an n x m matrix where n < m I would like to prove that there exists some [tex]\lambda[/tex] such that [tex]rank(A^T A + \lambda I) = m [/tex](adsbygoogle = window.adsbygoogle || []).push({});

I know that if two of the columns of [tex]A^T A[/tex] are linearly dependent, they are scalar multiples of each other and by adding some [tex]\lambda[/tex] to two different positions, those colums will become independent but I can't prove it for more than two columns.

Any tips?

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# Prove property of ranks

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