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Mathematics-Online problems:

Interactive Problem 309: States on Eigenvalues and Diagonalisable Matrices

A B C D E F G H I J K L M N O P Q R S T U V W X Y Z
Consider $ A\in\mathbb{C}^{n\times n}$ with $ n\in\mathbb{N}$. Let $ \lambda_1,\,\ldots , \lambda_k$ be the eigenvalues of $ A$ and $ d_1,\,\ldots , d_k$ the corresponding geometric multiplicities. Let $ E_n$ be the $ n\times n$-identity matrix. Mark which states are true respectively false:

$ A=\bar{A}$ and $ A$ symmetric $ \Longrightarrow$ $ A$ is diagonalisable n/a true false
$ d_1=\ldots = d_k=1$ $ \Longrightarrow$ $ A$ is diagonalisable n/a true false
$ A$ is diagonalisable $ \Longrightarrow$ $ {\rm {rank}}\,A=n$ n/a true false
$ A$ is diagonalisable $ \Longrightarrow$ $ A^4-A^3+E_n$ is diagonalisable n/a true false


   

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  automatically generated: 8/11/2017