# Inequality involving positive definite operator

• ismaili
In summary, the conversation is discussing how to prove the inequality |\langle f|H|g\rangle|^2 \leqslant \langle f|H|f\rangle \langle g|H|g\rangle, given the condition that |\langle f|g\rangle|^2 \leqslant \langle f|f\rangle\langle g|g\rangle and H is a Hermitian and positive definite operator. The participants suggest using the square root of H and proving that H satisfies the defining properties of an inner product in order to prove the inequality. Ultimately, it is determined that the problem is somewhat trivial because the Schwartz inequality automatically applies to inner products.
ismaili
1.
Given that $$|\langle f|g\rangle|^2 \leqslant \langle f|f\rangle\langle g|g\rangle$$
prove that $$|\langle f|H|g\rangle|^2 \leqslant \langle f|H|f\rangle \langle g|H|g\rangle$$
where $$H$$ is a Hermitian and positive definite operator.

3. I tried to identify $$H|g\rangle$$ as a state and put it into the given inequality, but not so help.

Is there any ideas to prove this inequality? Thanks in advance.

You will also need to think about the conditions H should meet for its square root to exist.

weejee said:

You will also need to think about the conditions H should meet for its square root to exist.

Thank you so much!
I got it, and in this way, Hermitian and positive definite properties are both used.
Thanks!

The problem is kind of silly to begin with. If you can show that

$$\langle \cdot | H | \cdot \rangle$$

satisfies the defining properties of an inner product:

1. Positive definite,

2. Linear on second slot,

3. Conjugate-linear on first slot,

then the expression automatically obeys the Schwartz inequality, because the Schwartz inequality is true of inner products in general.

Ben Niehoff said:
The problem is kind of silly to begin with. If you can show that

$$\langle \cdot | H | \cdot \rangle$$

satisfies the defining properties of an inner product:

1. Positive definite,

2. Linear on second slot,

3. Conjugate-linear on first slot,

then the expression automatically obeys the Schwartz inequality, because the Schwartz inequality is true of inner products in general.

Well, you are right. I keep forget that an inner product doesn't always have to be what we usually consider as 'the inner product'.

## 1. What is a positive definite operator?

A positive definite operator is a linear transformation on a vector space that satisfies certain conditions, including having all positive eigenvalues and producing a positive inner product when applied to any non-zero vector.

## 2. How is inequality involving positive definite operator useful in science?

Inequality involving positive definite operator is useful in science because it allows us to compare the properties of different objects or systems using mathematical tools. It can also help us understand the relationships between different variables and make predictions about their behavior.

## 3. Can you give an example of an inequality involving positive definite operator?

One example of an inequality involving positive definite operator is the Cauchy-Schwarz inequality, which states that for any two vectors x and y in a vector space, the inner product of x and y squared is less than or equal to the inner product of x with itself multiplied by the inner product of y with itself.

## 4. How is inequality involving positive definite operator related to matrix theory?

Inequality involving positive definite operator is closely related to matrix theory, as positive definite operators can be represented by symmetric positive definite matrices. Inequality involving positive definite operator is used in matrix theory to prove theorems and to analyze the properties of matrices.

## 5. What are some applications of inequality involving positive definite operator in real-world problems?

Inequality involving positive definite operator has many applications in real-world problems, such as in optimization, signal processing, and statistics. It is also used in machine learning and data analysis to compare the performance of different models and algorithms. Inequality involving positive definite operator is also important in physics, particularly in quantum mechanics, where it is used to describe the uncertainty principle.

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