I am working on the problem from the Charu C. Aggarwal's Linear Algebra book which ask to show that the Frobenius norm of the inverse of an $n × n$ matrix with Frobenius norm of $\epsilon$ is at least $\sqrt{\frac{n}{\epsilon}}$. I know that we define the Frobenius norm of a matrix as a square root of the sum of the squared elements of the matrix as follows:
$$\left\| A \right|_F = \sqrt{\sum_{i=1}^n\sum_{j=1}^na_{i,j}^2} = \epsilon$$
But I cannot find a way to connect above information with the Frobenius norm of the inverse.
I would be thankful for some hint how to proceed further with this task.
Best Answer
The norm is sometimes called the Hilbert-Schmidt norm. This norm is associated with the inner product of matrices as follows $$\langle A,B\rangle =\sum_i\sum_j a_{ij} \overline{b_{ij}}={\rm Tr}( B^*A).$$ Observe that $$\langle A,B\rangle =\langle I, A^*B\rangle.$$ Now $$n=\langle I,I\rangle =\langle I, A\, A^{-1}\rangle =\langle A^*,A^{-1}\rangle.$$ The last expression can be estimated above using Cauchy-Schwarz inequality, by the product of norms. Hence $$\|A^{-1}\|\ge {n\over \|A\|}.$$