Inconsistent Inner Product Definitions

[cepheid]

Hi,

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

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 \mathbf{x} is replaced by \mathbf{\bar{x}}, and the inner product of \mathbf{x} and \mathbf{y} becomes:

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

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?" )

\mathbf{\bar{x}}^{\mathrm{T}} = \mathbf{x}^{\mathrm{H}}

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

From Elementary Differential Equations and Boundary Value Problems by Boyce and DiPrima, 8th ed., pg. 397:

...the scalar or inner product [...] is defined by

(\mathbf{x}, \mathbf{y}) = \mathbf{x}^{\mathrm{T}} \mathbf{\bar{y}} = \sum_{i=1}^n x_i \bar{y}_i

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 Introduction to Quantum Mechanics in 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.

 

[Hurkyl]

Both conventions are used. I think the "physicist" convention is antilinear in the first argument, and the "mathematician" convention is antilinear in the second argument.