Discussion Overview
The discussion revolves around the multiplication of column vectors, specifically examining the relationship between the inner product and outer product of a column vector and its conjugate transpose. The scope includes theoretical aspects of linear algebra and vector operations.
Discussion Character
- Technical explanation
- Conceptual clarification
Main Points Raised
- One participant questions whether the equality \(\mathbf{h}^*\mathbf{h}=\mathbf{h}\mathbf{h}^*\) holds for an \(N \times 1\) column vector \(\mathbf{h}\).
- Another participant suggests testing the equality with a specific example, \(\mathbf{h} = [1, 0]^*\), and mentions the concepts of inner product and outer product as relevant operations.
- It is noted that \(\mathbf{h}\mathbf{h}^*\) represents the outer product and \(\mathbf{h}^*\mathbf{h}\) represents the inner product, with a clarification that the inner product is typically a real number rather than a \(1 \times 1\) matrix.
- Further discussion highlights the applications of the outer product in dual vector spaces and its relevance in quantum mechanics (QM) and general relativity (GR).
Areas of Agreement / Disagreement
Participants generally agree on the definitions and applications of inner and outer products, but the initial question regarding the equality of the two expressions remains unresolved.
Contextual Notes
The discussion does not resolve whether the equality holds under all conditions, and there may be assumptions regarding the properties of the vectors involved that are not explicitly stated.