# Integration of traceless symmetric matrices

• jouvelot
In summary: Your Name]In summary, the conversation discusses a formula discovered by Pierre regarding tensor perturbations in cosmology. The formula involves a symmetric traceless matrix-valued function of a unit vector, and a function of the scalar product between the unit vector and another vector. The formula can be simplified by using the fact that the unit vector is perpendicular to the other vector in a two-dimensional space. This simplification does not require the use of spherical or euclidean coordinate expressions.
jouvelot
Hi,

I stumbled upon an identity when studying tensor perturbations in cosmology. The formula states that
$$\int d^2\hat{p} f(\hat{p}.\hat{q})\hat{p}_i \hat{p}_k e_{jk}(\hat{q}) = e_{ij}(\hat{q})/2 \int d^2\hat{p} f(\hat{p}.\hat{q})(1-(\hat{p}.\hat{q})^2),$$ where ##e_{ij}## is a symmetric traceless matrix-valued function of ##\hat{q}## with ##\hat{q}_i e_{ij}(\hat{q}) = 0##, ##\hat{p}## is the unit vector parallel to ##p## and ##f## any function of the scalar product ##\hat{p}.\hat{q}##.

Is there a simpler way to get this formula than to proceed by using cumbersome spherical or euclidian coordinate expressions?

Bye,

Pierre

Last edited:

Dear Pierre,

Thank you for sharing your discovery with us. The formula you have stumbled upon is indeed quite interesting and can provide valuable insights in the study of tensor perturbations in cosmology. I have looked into it and have found a simpler way to arrive at this formula.

Instead of using spherical or euclidean coordinate expressions, we can use the fact that the unit vector ##\hat{p}## is perpendicular to ##\hat{q}## in a two-dimensional space. This means that the dot product ##\hat{p}.\hat{q}## is equal to the sine of the angle between the two vectors.

Using this, we can rewrite the formula as:
$$\int d^2\hat{p} f(\sin\theta)\hat{p}_i \hat{p}_k e_{jk}(\cos\theta) = e_{ij}(\cos\theta)/2 \int d^2\hat{p} f(\sin\theta)(1-\cos^2\theta),$$
where ##\theta## is the angle between ##\hat{p}## and ##\hat{q}##.

This form of the formula is much simpler and can be easily understood without the need for cumbersome coordinate expressions. I hope this helps in your further study of tensor perturbations in cosmology.

## 1. What is the definition of a traceless symmetric matrix?

A traceless symmetric matrix is a square matrix where the sum of the elements on the main diagonal (known as the trace) is equal to zero, and the elements are symmetric about the main diagonal.

## 2. How is integration of traceless symmetric matrices useful in scientific research?

Integration of traceless symmetric matrices is useful in various fields of science, such as physics, engineering, and computer science. It is used to solve problems involving vector calculus, differential equations, and optimization.

## 3. What are some properties of traceless symmetric matrices?

Some properties of traceless symmetric matrices include:

• The sum of the eigenvalues is equal to zero.
• The determinant is equal to the product of the eigenvalues.
• The inverse of a traceless symmetric matrix is also a traceless symmetric matrix.

## 4. How is the integration of traceless symmetric matrices performed?

The integration of traceless symmetric matrices is typically done using numerical methods, such as the trapezoidal rule or Simpson's rule. These methods involve dividing the matrix into smaller parts and approximating the integral using the sum of these smaller parts.

## 5. Can traceless symmetric matrices be used in real-world applications?

Yes, traceless symmetric matrices have various real-world applications, such as in quantum mechanics, image processing, and machine learning. They are also used in the study of physical systems, such as fluid dynamics and elasticity.

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