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Keeping in mind that the set of all n $\times$ n matrices over a field F forms a vector space isomorphic to L(V, V) where dim V = n & all the result of eigen values of a square matrix has its analogous outcomes for corresponding linear operator, can we transform any problems involving the eigen values of square matrix to a analogous problem for operator. Let us take an example: Let T ($\neq$ $O$) be a linear operator on a n-dimensional real vector space V such that Rank (T) = k < n. Suppose for some real $\lambda$, $T^2$=$\lambda$$T$. Then which of followings are true: 1. $\lambda$=1, 2. det T = $|\lambda|^n$, 3. $\lambda$ is the only eigen value of T, 4. $\exists$ a non-trivial subspace $V_0$$\subset V$ sucht that $Tx=0$ $\forall$ $x \in V_0$. If 2 & 4 are the correct alternatives of the problem then can we say 2 & 4 are also the correct alternatives of the corresponding problem obtained by replacing the term 'linear operator' with 'martix' & vice-versa.

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Note that it is possible to have $T^2=0$ without having $T=0$. Among other things, this implies $\lambda$ could be zero. –  Gerry Myerson Dec 6 '12 at 5:22
    
@GerryMyerson:Would someone of you please checkout whether my approach to the given problem is right: (4) is true: $Nullity(T)+Rank(T)=n$ $\Rightarrow$ $Nullity(T) = n - k>0$ $\Rightarrow$ $Ker$ $T \neq 0$ $\Rightarrow$ (4) is true; (1) is false: Consider $T:V_n\to V_n:v\mapsto -v$$\Rightarrow$ $T^2=(-1)T$ $\Rightarrow$ (1) is false; (3) is false: Let T = $$\begin{pmatrix}1&0\\0&0\\\end{pmatrix}$$ Then $T$ satisfies the given conditions but here $\lambda = 1$ is not the only eigenvalue of $T$; (2) is false: The above example shows that det $T$ = $0\neq |1|^2$. –  Sugata Adhya Dec 7 '12 at 10:40
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Looks good to me. –  Gerry Myerson Dec 7 '12 at 11:42
    
Thanks @GerryMyerson –  Sugata Adhya Dec 7 '12 at 11:48
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up vote 2 down vote accepted

Yes, in your example. In general, some care must be taken. For example, any matrix is similar to its Jordan form, and this is useful; but there are properties of operators that are not preserved by similarity, like positivity and any orthogonal-related property.

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