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I'm looking at the definition of the inner product of two vectors in [itex] \mathbb{C}^n [/itex]. One source is talking about how the definition of an inner product must be modified to account for vectors with complex components and says:

He then goes on to say that we can rewrite conjugate transpose as follows: (a.k.a. the hermitian conjugate or hermitian transpose, depending which book you read, it seems. Can't we just stick to "adjoint?" ) From Linear Algebra and its Applications by Gilbert Strang, 3rd ed., pg. 293:

... the standard modification is to conjugate the first vector in the inner product. This means that [itex] \mathbf{x} [/itex] is replaced by [itex] \mathbf{\bar{x}} [/itex], and the inner product of [itex] \mathbf{x} [/itex] and [itex] \mathbf{y} [/itex] becomes:

[tex] \mathbf{\bar{x}}^{\mathrm{T}} \mathbf{y} = \bar{x}_1 y_1 + \bar{x}_2 y_2 + \cdots + \bar{x}_n y_n [/tex]

[tex] \mathbf{\bar{x}}^{\mathrm{T}} = \mathbf{x}^{\mathrm{H}} [/tex]

The point of this thread is that I have a second source with a contradictory definition (the second vector conjugated instead of the first):

So what gives? Which is the correct definition? I'm inclined to believe the first one (G. Strang) if only because it is consistent with the definition given by Griffiths in his From Elementary Differential Equations and Boundary Value Problems by Boyce and DiPrima, 8th ed., pg. 397:

...thescalarorinner product[...] is defined by

[tex] (\mathbf{x}, \mathbf{y}) = \mathbf{x}^{\mathrm{T}} \mathbf{\bar{y}} = \sum_{i=1}^n x_i \bar{y}_i [/tex]Introduction to Quantum Mechanicsin Appendix A. So that's 2 sources vs. 1. Griffiths of course, uses the wacky physics notation <a|b>, which I'm still not totally used to. He also uses totally different notation for complex conjugation and the transpose, and the adjoint.

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# Inconsistent Inner Product Definitions

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