MathDB

6

Part of 2002 IMC

Problems(1)

IMC 2002 Problem 6

Source: IMC 2002

3/7/2021
For an n×nn\times n matrix with real entries let ∣∣M∣∣=sup⁡x∈Rn∖{0}∣∣Mx∣∣2∣∣x∣∣2||M||=\sup_{x\in \mathbb{R}^{n}\setminus\{0\}}\frac{||Mx||_{2}}{||x||_{2}}, where ∣∣⋅∣∣2||\cdot||_{2} denotes the Euclidean norm on Rn\mathbb{R}^{n}. Assume that an n×nn\times n matrxi AA with real entries satisfies ∣∣Ak−Ak−1∣∣≤12002k||A^{k}-A^{k-1}||\leq\frac{1}{2002k} for all positive integers kk. Prove that ∣∣Ak∣∣≤2002||A^{k}||\leq 2002 for all positive integers kk.
linear algebramatrix