I'm stuck to prove Nilpotent Matrix

In summary, a matrix is considered nilpotent if there exists a positive integer k such that A^k = O, where A is the matrix and O is the zero matrix. Nilpotent matrices play an important role in various areas of mathematics and can be proven using the Cayley-Hamilton theorem. Non-square matrices cannot be nilpotent and all nilpotent matrices are singular, meaning they do not have an inverse.
  • #1
crazygrey
7
0
Hi all,
If a square matrix A of dimension n*n is a nilpotent matrix,i.e, A^k=0 for k>=m iff A has eigenvalues 0 with multiplicity n and index m or less. I did prove by induction if A^k=0 then all the eigenvalues are zero. I'm lost when I want to prove the oppesite, i.e, if all eigenvalues are zero then A^k=0? Please help
 
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  • #2
Put it in jordan normal form and it all drops out. Alternatively just think about the characteristic poly.
 

What does it mean for a matrix to be nilpotent?

A matrix is considered nilpotent if there exists a positive integer k such that Ak = O, where A is the matrix and O is the zero matrix. This means that when the matrix is raised to the kth power, it results in a matrix where all the elements are equal to zero.

What is the significance of a nilpotent matrix in mathematics?

Nilpotent matrices play an important role in various areas of mathematics, such as linear algebra and group theory. They are particularly useful in studying the structure of Lie algebras and their representations.

How can I prove that a matrix is nilpotent?

To prove that a matrix is nilpotent, you can use the Cayley-Hamilton theorem, which states that every square matrix satisfies its own characteristic polynomial. By substituting the matrix into its characteristic polynomial and simplifying, you can show that it equals the zero matrix, thus proving nilpotency.

Can a non-square matrix be nilpotent?

No, a non-square matrix cannot be nilpotent. This is because the definition of a nilpotent matrix requires it to be raised to a positive integer power, which is not possible for a non-square matrix since the number of columns must equal the number of rows to perform matrix multiplication.

Are all nilpotent matrices singular?

Yes, all nilpotent matrices are singular, meaning they do not have an inverse. This is because the determinant of a nilpotent matrix is always equal to zero, making it impossible to find an inverse using the standard method of matrix inversion.

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