Explicit expression for inverse of I-A

In summary, The conversation is about finding an explicit expression for (I-A)^(-1), given that A is a squared matrix (nxn) and A^k = 0. It is also mentioned that I-A^k = (I-A)(I+A+A^2+...+A^(k-1)). The speaker is looking for a simplified and explicit expression for (I-A)^(-1) without knowing what A is. The other person suggests that I+A+A^2+...+A^(k-1) is an explicit expression, but the speaker is still looking for a more simplified one.
  • #1
peripatein
880
0
Hello,
This is not a homework exercise, so I decided to post it here. Hopefully one of you could help.
I would like to find an explicit expression for (I-A)^(-1), provided that A is a squared matrix (nxn) and A^k = 0. It is also given that I-A^k = (I-A)(I+A+A^2+...+A^(k-1)).
I understand that by definition the inverse matrix of I-A will be (I+A+A^2+...+A^(k-1)), but is there a way to arrive at a more simplified, explicit expression (yet without knowing what A is)?
 
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  • #2
I don't understand why

[tex]I+A+A^2+...+A^{k-1}[/tex]

isn't good enough for you. It's an explicit expression.

To my knowledge, there is no other expression.
 

1. What is the definition of "Explicit expression for inverse of I-A"?

The explicit expression for inverse of I-A refers to the mathematical formula that can be used to calculate the inverse of a matrix (A) minus the identity matrix (I).

2. Why is the explicit expression for inverse of I-A important?

This expression is important because it allows us to efficiently calculate the inverse of a matrix without having to use time-consuming methods such as Gauss-Jordan elimination or finding the determinant.

3. How is the explicit expression for inverse of I-A derived?

The explicit expression for inverse of I-A is derived using the Cayley-Hamilton theorem, which states that every matrix satisfies its own characteristic equation. By manipulating this characteristic equation, we can obtain the explicit expression for the inverse of I-A.

4. Can the explicit expression for inverse of I-A be used for any matrix?

No, the explicit expression for inverse of I-A can only be used for square matrices (nxn) that are invertible, meaning they have a non-zero determinant.

5. Are there any limitations to using the explicit expression for inverse of I-A?

One limitation is that the explicit expression may not be valid if the matrix (A) is singular, meaning it does not have an inverse. Additionally, the calculation may become computationally expensive for large matrices due to the use of higher order determinants.

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