Given the matrix

2nd is aslo correct

let me check it

Yes I updated my answer

The result that I have wrote for such a matrix M is very helpful.

can you please explain what am & gm is...

i am getting a bit confusion there

see

Yes 2nd is also correct

let me explain u am and gm

However, GM of the Eigen value say "d" is the dimension of the null space of the matrix A- dI, where I is identity matrix

Am of Eigen value is its multiplicity as a root of the characteristics polynomial of the matrix

Harshal, if you still have any doubt....I will surely try to clear it...you can ask without hesitation

what is AM and GM?

and it is msq type que

AM - algebraic multiplicity GM = geometric multiplicity

this result is true only for real symmetric matrix, not over complex field

use the result for finding eigen value ..which is -i and 3 i

then since rank is 4, there will be 4 eigenvalues and rank of this matrix is 4 , it is diagonalizable also (AM=GM) ..

read about AM any GM ..to know more..

any doubt then ask..

u said rank = 4, so it must have 4 Eigen values..... but Eigen values has nothing to do with rank of a matrix.... for example rotational matrix has rank 2 but it has no Eigen values

rank of matrix = no. of eigenvalue

there will be 4 eigenvalues , no doubt ..

I didn't understand ur reason for diagonalizability

read this

read it carefully...these statements are true for real symmetric only

dear shashi , diagonalizable iff AM=GM holds for both complex and real..