I need help understanding and solving this problem.
Prove or give a counterexample: If A is a nonsingular matrix, then ‖A−1‖=‖A‖−1
Is this just asking me to get the magnitude of the inverse of Matrix A, and then compare it with the inverse of the magnitude of Matrix A?
Answer
If A is nonsingular, then AA−1=I, so
1=||I||=||AA−1||⩽
In general, then 1 \leqslant ||A||\cdot||A^{-1}|| \implies ||A||^{-1} \leqslant ||A^{-1}||.
Equality is thus not necessarily guaranteed for arbitrary nonsingular A; however, the inequality above implies that equality may occur. Consider an example.
Example:
A = \begin{bmatrix} 1 & 0 \\ 0 & 2 \end{bmatrix}, A^{-1} = \begin{bmatrix} 1 & 0 \\ 0 & 0.5 \end{bmatrix}
\implies ||A||_{1,2,\infty} = 2
\implies ||A^{-1}||_{1,2,\infty} = 1
\implies \frac{1}{2} = ||A||_{1,2,\infty}^{-1} \neq ||A^{-1}||_{1,2,\infty} = 1 \implies ||A||^{-1} \neq ||A^{-1}||.
Attribution
Source : Link , Question Author : Gary , Answer Author : clocktower
