# Deriving Nilpotent Matrices: I+N^-1 = I - N + N^2 - N^3...

• brydustin
In summary, the equation (I+N)^-1 = I - N + N^2 - N^3 + ... N^(k-1) + 0 can be derived by multiplying (I+N) with (I-N+N^2-N^3+...N^{k-1}). This results in the "middle parts" cancelling out and leaving only the Identity matrix, showing that the inverse exists. This equation is derived assuming the existence of the inverse.

#### brydustin

I am curious how to derive the (I+N)^-1 = I - N + N^2 - N^3 + ... N^(k-1) + 0
Where N^k = O, because we assume that N is nilpotent.

Actually I'm just supposed to show that the inverse always exists (for my homework), but I'm not asking how to find existence, I want to know how this equation is derived (assuming existence).

Thanks...

What happens if you multiply

$$(I+N)(I-N+N^2-N^3+...N^{k-1})$$

AH! Thanks... all the "middle parts fall out

I - N + N - N^2 + N ^2 -... - N^(k-1) +N^(k+1) + N^k

and you are left with Identity, which shows that its the inverse. Thanks :)

## 1. What is a nilpotent matrix?

A nilpotent matrix is a square matrix that, when raised to a certain power, becomes the zero matrix. This means that all of its entries become zero after a certain number of multiplications.

## 2. How is a nilpotent matrix derived?

A nilpotent matrix can be derived using the formula I + N^-1 = I - N + N^2 - N^3 + ..., where I is the identity matrix and N is the nilpotent matrix. This formula is based on the geometric series formula, where the sum of an infinite geometric series is equal to the first term divided by one minus the common ratio.

## 3. What is the significance of nilpotent matrices?

Nilpotent matrices are important in linear algebra and in the study of systems of linear equations. They can be used to find the solutions of homogeneous systems of linear equations, and they are also used in the Jordan canonical form of a matrix.

## 4. Can a nilpotent matrix have non-zero entries?

Yes, a nilpotent matrix can have non-zero entries. The only requirement for a matrix to be nilpotent is that it becomes the zero matrix after a certain number of multiplications, regardless of the initial values of its entries.

## 5. Are all nilpotent matrices similar?

No, not all nilpotent matrices are similar. Two matrices are similar if they have the same eigenvalues and eigenvectors. However, a nilpotent matrix can have different eigenvalues and eigenvectors depending on its size and the values of its entries, so not all nilpotent matrices are similar.