how do i find the index of a nilpotent matrix??..
Hmm. I don't know of any "formula" as such. You know that an NxN matrix will have index <= N, so just start multiplying. You'll have at most N-1 multiplications.
but what if N is a huge number...is there any logic to make the process simpler?
To find the index of a nilpotent matrix, you need to perform the following steps:
Step 1: Determine if your matrix is nilpotent. A square matrix A is nilpotent if there exists a positive integer k such that A^k = O, where O is the zero matrix.
Step 2: Begin by checking if A^2 is the zero matrix. If it is, then the matrix A is of index 2. Otherwise, continue to the next step.
Step 3: Compute A^3 and check if it is the zero matrix. If it is, then the matrix A is of index 3. Otherwise, proceed to the next step.
Step 4: Repeat this process by computing A^4, A^5 and so on until you find the smallest value of k such that A^k = O. This value will be the index of the nilpotent matrix A.
Note: Calculating higher powers of a matrix can be done using matrix multiplication. Multiply the matrix A by itself repeatedly until you reach the desired power.
If you are using computer software such as MATLAB or Python, you can simply raise the matrix to the desired power and check if it is the zero matrix. This will expedite the process of finding the index of the matrix.