danik_ejik said:I tried multiplying (I + A) and (I - A/2) and indeed it equals I. That's it!? That proves it ?
To prove the invertibility of a matrix I + A, you can use the fact that for a projection matrix A, (I + A) is also a projection matrix. This means that (I + A)^2 = (I + A), which implies that (I + A) is its own inverse. Therefore, (I + A) is invertible and its inverse is (I + A) itself.
A projection matrix is a square matrix that, when multiplied by itself, gives the same matrix. This means that it projects a vector onto a subspace, hence the name "projection" matrix. It has the properties of being symmetric, idempotent, and having eigenvalues of 0 or 1.
Proving the invertibility of a matrix I + A for projection matrices A is important because it shows that the matrix is well-defined and has a unique solution. It also allows for easier calculations and simplification of equations when working with projection matrices.
Yes, a projection matrix can be singular. If a projection matrix has eigenvalues of 0 or 1, it will be singular.
The invertibility of a matrix I + A for projection matrices A is related to linear transformations because projection matrices represent linear transformations that project vectors onto a subspace. An invertible projection matrix ensures that there is a unique transformation from the subspace back to the original vector space.