VI.6) (*) Determine if the following are true or false and...

VI.6) (*) Determine if the following are true or false and think of a brief explanation of why that is the case. (i) Given $S, T \in \mathcal{L}(V)$ two nilpotent operators, then $ST$ is nilpotent. (ii) Let $T \in \mathcal{L}(V)$ be nilpotent and diagonalizable, then $T = 0$. (iii) Given $S, T \in \mathcal{L}(V)$ two nilpotent operators, then $S + T$ is nilpotent. (iv) Given $S, T \in \mathcal{L}(V)$ two nilpotent operators such that $ST = TS$, then $S + T$ and $ST$ are nilpotent. (v) Let $T \in \mathcal{L}(V)$, assume that there exists $B_V$ a basis of $V$ such that $\mathcal{M}(T, B_V)$ is a diagonal matrix. Then for any Jordan basis $B_V'$ the matrix $\mathcal{M}(T, B_V')$ is diagonal. (vi) Let $T \in \mathcal{L}(V)$ on a complex vector space and consider $B_V$ and $B_V'$ two different Jordan basis. Then $\mathcal{M}(T, B_V)$ and $\mathcal{M}(T, B_V')$ can have a different number of blocks in its diagonal form.

Transcript

0:00 Hello students.

00:02 So from the above question it is true or false so we will see one by one.

00:08 So first we'll see the a part so the a part answer is true.

00:14 So why it is true means so why it is true means a square matrix.

00:20 A square matrix the square matrix is set to be diagonalized is said to be is set to be diagonalized it is diagonalizable if it is similar to the diagonal matrix if it is similar to diagonal matrix so what is diagonal matrix you all know so 1 -0 -0 -0 -0 -0 and 0 -0 -0 -1 so these are called as the diagonal matrix okay when a square matrix is said to be diagonalized if it is similar to the diagonal matrix when it is in the diagonal matrix only it is it will be diagonalized so the first one answer is true okay so second one b so b the answer is false okay so why the answer is false means diagonalizability does not supply invertibility so the meaning will be diagonality diagonalizability does not sorry does not implies okay diagonal as well does not imply invertibility so diagnosis means in the above i have explained diagonal means 1 -0 -0 -0 -0 -0 -0 -0 -0 -0 -1 so these are diagonality inverse means a inverse of something okay so here the matrix itself will be changed so this is inverse and this is diagonal so diagonal and inverse are totally different thing so it does not imply diagonal does not imply invertibility so the second option b is false okay next the third one see the third one is true okay so why it is true means a n cross n a n cross n matrix with n different with n different eigenvalue so with with n different eigenvalue so with with n different eigenvalue must have a basics of eigenvector must have basic of eigenvector.

03:15 So here the eigenvalues must have the eigenvector value.

03:20 So when it has the eigenvector value means only it will be it it implies for this end cross end matrix.

03:28 So otherwise it will not imply for an n cross end matrix.

03:31 So the eigenvalue must have a eigenvector.

03:35 So the option c is true.

03:39 So the last one, sorry, the d one is true.

03:44 Okay, the next one answer is true.

03:48 So why it is true means if a is equal to bd b vector, sorry, b inverse, then d is equal to b a b inverse...

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