Which one of the following are correct?[CSIR-Dec 2018]

Asked Jun 15 '19 at 01:42

Active Jun 15 '19 at 06:34

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let $M$ be a hermitian matrix of order $n*n$ with rank k$ \neq n$

My attempt:-

I know that $u^*u=1$. Since eigenvectors are orthonormal set for a Hermitian matrix. I tried to solve using rank$(M(I-\lambda uu^*))\leq \min\{rank(M),rank(I-\lambda uu^*)\}$ for (1) and (2) So, option 3 is obviously false. Since, rank$(M(I-\lambda uu^*))\leq \min\{rank(M),rank(I-\lambda uu^*)\}\leq k$ since Rank $M\leq k$ (4) Option is true, since by Induction, I got it is correct. How do I solve (1) and (2)?

edited Jun 15 '19 at 01:49

asked Jun 15 '19 at 01:42

Can you write the eigendecomposition for $M$? – Raghav Jun 15 '19 at 02:12
What do you mean by eigen decomposition? – Jun 15 '19 at 02:16
Spectral theorem allows one to write a Hermitian matrix $M$ as $M=\sum \lambda_i u_i u_i^*$ where the sum is over non-zero eigenvalues of $M$ and $u_i$ is a unit eigenvector corresponding to the eigenvalue $\lambda_i$. – Raghav Jun 15 '19 at 02:23
Why are you posting an answer to an m.se question as an m.se question? – Gerry Myerson Jun 15 '19 at 03:08
1,2,3 will not be true in all situations since they depend on the geometric multiplicity of the eigenvalue $\lambda$ – Ben Grossmann Jun 15 '19 at 03:33
@Gerry the answer was posted after this question, apparently. The question remains however as to why he posted another question as a question, though. – Ben Grossmann Jun 15 '19 at 03:34
hello, maneesh, please visit the chat, I have pinged you – vidyarthi Jun 27 '19 at 17:54

Which one of the following are correct?[CSIR-Dec 2018]

0 Answers0