The discussion revolves around the properties of skew symmetric matrices, specifically focusing on the invertibility of the matrix formed by adding a skew symmetric matrix \( A \) to the identity matrix \( I \). The original poster presents a problem involving proving that \( I + A \) is invertible, following an initial proof related to the quadratic form \( u^T A u = 0 \) for any vector \( u \).
There is an ongoing exploration of the properties of the matrix \( I + A \) and its kernel. Some participants have offered hints and guidance, while others express uncertainty about the concepts being discussed. The conversation reflects a mix of understanding and confusion, with no explicit consensus reached yet.
Participants note a lack of familiarity with certain linear algebra concepts, such as the kernel and injective operators, which may be affecting their ability to engage fully with the problem. There is also an acknowledgment of previous discussions on similar topics, indicating a shared learning environment.
micromass said:
micromass said:
The kernel is exactly thesame as the null space. And injective (or one-to-one or monomorphism) just means that the null space is trivial.