# Inner product - positive or positive semidefinite?

• I
• dyn
In summary: Yes, but you are not listening to what I am saying. His inner product is positive definite, he includes the condition ##\langle v|v\rangle = 0## iff ##v=0##. He just calls it positive semi-definite.
dyn
Hi

In QM the inner product satisfies < a | a > ≥ 0 with equality if and only if a = 0.

Is this positive definite or positive semidefinite because i have seen it described as both

Thanks

Inner products ##<a|b> ## is a complex number in general.
Inner product with its conjugate ##<a|a>## is positive and usually to be normalized. I am not sure
whether we include ##|a>=0## as a state vector which allows semidefinite nature.

vanhees71

vanhees71
In the standard definition of a Hilbert space the scalar product is positive definite, i.e., if ##\langle a|a \rangle=0##, the vector ##|a \rangle=|0 \rangle## is the null vector.

That's why, e.g., in its realization as the Hilbert space of square integrable functions (as used in "wave mechanics" a la Schrödinger) a function represents the null vector if it is zero except on a zero-measure null [edit: corrected in view of #5; and once more in view of #7] set.

Last edited:
anuttarasammyak
vanhees71 said:
In the standard definition of a Hilbert space the scalar product is positive definite, i.e., if ##\langle a|a \rangle=0##, the vector ##|a \rangle=|0 \rangle## is the null vector.

That's why, e.g., in its realization as the Hilbert space of square integrable functions (as used in "wave mechanics" a la Schrödinger) a function represents the null vector if it is zero except on a zero set.
You mean except on a zero measure set.

vanhees71
Thanks, I wasn't sure about the correct english term for "Nullmenge" in the sense of Lebesgue-integration theory.

vanhees71 said:
Thanks, I wasn't sure about the correct english term for "Nullmenge" in the sense of Lebesgue-integration theory.
Ah, i see! It is null set.

vanhees71
So much for the reliability of online dictionaries :-(. BTW does anybody know a good one for mathematics/physics vocabulary? One translated it with "empty set", which doesn't make sense in this context or with "zero set" which may be wrong either.

Demystifier said:
The definition given here shows that in terms of QM the inner product can be defined as positive definite or positive semi-definite which might account for the fact that Shankar calls it positive semidefinite while Sakurai calls it positive definite

dyn said:
The definition given here shows that in terms of QM the inner product can be defined as positive definite or positive semi-definite which might account for the fact that Shankar calls it positive semidefinite while Sakurai calls it positive definite
You mix the positive definiteness of the scalar product of a Hilbert space, and by definition a Hilbert space has a positive definite scalar product and the positive definiteness of a self-adjoint operator.

vanhees71 said:
You mix the positive definiteness of the scalar product of a Hilbert space, and by definition a Hilbert space has a positive definite scalar product and the positive definiteness of a self-adjoint operator.
I looked at Shankar. He does call it semidefinite.

What does Shankar call semidefinite? The scalar product? I'd be surprised!

vanhees71 said:
What does Shankar call semidefinite? The scalar product? I'd be surprised!
Yes.

That's very strange. Then QT doesn't work the usual way!

vanhees71 said:
That's very strange. Then QT doesn't work the usual way!
No, his inner product is the same, he just calls it semidefinite.

Ok, an inner product is positive semidefinite if for all vectors ##|v \rangle## you have ##\langle v|v \rangle \geq 0##. Take a finite-dimensional space. Then you can choose a basis such that
$$\langle v|v \rangle=\sum_{j=1}^d g_{jk} v_j^* v_k,$$
and you must have ##(g_{jk})=\mathrm{diag}(1,1,\ldots,1,0,0,0\ldots)##. In that case the inner product does not induce a metric.

That's why, for a scalar product you define it to be positive definite, i.e., it's positive semidefinite and in addition you have ##\langle v|v \rangle = 0 \Leftrightarrow |v \rangle=|0 \rangle## (where ##|0 \rangle## is the null vector).

vanhees71 said:
Ok, an inner product is positive semidefinite if for all vectors ##|v \rangle## you have ##\langle v|v \rangle \geq 0##. Take a finite-dimensional space. Then you can choose a basis such that
$$\langle v|v \rangle=\sum_{j=1}^d g_{jk} v_j^* v_k,$$
and you must have ##(g_{jk})=\mathrm{diag}(1,1,\ldots,1,0,0,0\ldots)##. In that case the inner product does not induce a metric.

That's why, for a scalar product you define it to be positive definite, i.e., it's positive semidefinite and in addition you have ##\langle v|v \rangle = 0 \Leftrightarrow |v \rangle=|0 \rangle## (where ##|0 \rangle## is the null vector).
Yes, but you are not listening to what I am saying. His inner product is positive definite, he includes the condition ##\langle v|v\rangle = 0## iff ##v=0##. He just calls it positive semi-definite.

So it's positive definite. That seems to be a pretty confusing textbook then.

gentzen
The Mathematical methods book by Riley & Hobson also states that positive semi-definite is < a | a > ≥ 0 with < a | a > = 0 implying a = 0

Ok, what then for those authors is "positive definite"? In the usual textbooks you have ##\langle a|a \rangle \geq 0## as the condition for a "positive semidefinite" sesquilinear form. If in addition ##\langle a| a \rangle =0 \Leftrightarrow |a \rangle=|0 \rangle## is fulfilled the sequilinear form is called "positive definite".

As usual, standard conventions followed by the vast majority of textbooks and scientific papers, make more sense than private nomenclature of some singular authors ;-).

DrClaude and gentzen
It seems that some physics textbooks can be less that strictly accurate with their mathematical terminology !

## 1. What is an inner product?

An inner product is a mathematical operation that takes in two vectors and produces a scalar value. It is commonly denoted as <x, y> and is used to measure the angle between two vectors, as well as the length of a vector.

## 2. What does it mean for an inner product to be positive?

A positive inner product means that the result of the operation is always a positive scalar value. This indicates that the two vectors are pointing in the same direction or have a positive correlation.

## 3. Can an inner product be negative?

Yes, an inner product can be negative. This means that the two vectors are pointing in opposite directions or have a negative correlation.

## 4. What is a positive semidefinite inner product?

A positive semidefinite inner product is one in which the result of the operation is always equal to or greater than zero. This indicates that the two vectors are either pointing in the same direction or are orthogonal (perpendicular) to each other.

## 5. How is the positive definiteness of an inner product important in mathematics?

The positive definiteness of an inner product is important in many areas of mathematics, including linear algebra, functional analysis, and optimization. It allows for the definition of important concepts such as norms, angles, and distances between vectors, which are crucial in various applications.

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