Prove that $\left(1 + \sum_{i=1}^{n}\left\|a_i\right\|_2^2 \right) < \det(I_m + AA^*) , A \in M_{m,n}$ and $a_i$ is $i$th column of $A$.

Question

• Prove the following inequality is true $$\left(1 + \sum_{i=1}^{n}\left\|a_i\right\|_2^2 \right) < \det(I_m + AA^*) \ ,$$

in which $A \in M_{m,n}$ and $a_i$ corresponds to the $i$th column of $A$.

• Also, prove that if $A$ is rank $1$ matrix, then the above inequality is an equality.

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I have tested this numerically (e.g., MATLAB) and it seems to be true. However, I am not sure how to prove it mathematically yet. Any suggestions and help will be highly appreciated.

I think it is closely related to this one: $\det\left(I + A^TA^{-1}\right) = 2\left(1 + \operatorname{tr}\left(A^TA^{-1}\right)\right)$ , but not same.

Answer 1

Let $M$ be a positive semi-definite matrix of size $m\times m$ and let $\lambda_1, \cdots, \lambda_m$ be its eigenvalues. Then

$$ \det(I_m+M) = \prod_{k=1}^{m} (1 + \lambda_k) \geq 1+\sum_{k=1}^{m} \lambda_k = 1+\operatorname{tr}(M). $$

Now the result follows by noticing that $\operatorname{tr}(AA^*) = \sum_{i=1}^{m}\sum_{j=1}^{n} \lvert a_{ij} \rvert^2$.