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.

 

[tacman]

First of all, it is not uncommon to encounter different definitions or notations for the same mathematical concept. However, in the case of inner products, there is a standard definition that is widely accepted and used in mathematics and physics. The first source you mentioned, Linear Algebra and its Applications by Gilbert Strang, follows this standard definition by conjugating the first vector in the inner product. This is known as the Hermitian inner product and is used in complex vector spaces. On the other hand, the second source, Elementary Differential Equations and Boundary Value Problems by Boyce and DiPrima, uses a slightly different definition by conjugating the second vector. This is known as the anti-Hermitian inner product and is used in some engineering and physics applications.

So, which one is the correct definition? It depends on the context and the application. In pure mathematics, the Hermitian inner product is the standard definition and is used in complex vector spaces. In physics and engineering, both definitions are used, depending on the specific problem or application. It is important to note that both definitions are equivalent and can be converted into each other by taking the complex conjugate of one of the vectors in the inner product.

As for the use of different notations, it is a matter of personal preference and does not affect the underlying mathematical concept. It is important to understand the definition and properties of an inner product, rather than getting caught up in notational differences. In summary, both definitions are valid and widely used, but it is important to understand the context and application in which they are being used.