# Designating matrices by (system2 operator system1)

• Lojzek
In summary: Your Name]In summary, the designation (S2 O S1) is commonly used in crystallography to represent matrices and is also known as the "coordinate-free" or "Einstein" notation. It is not limited to crystallography and is also used in other fields such as mathematics, physics, and engineering. This notation is helpful in keeping track of transformations between multiple coordinate systems. For further information, please refer to the links provided.
Lojzek
Hi,

I read a text about crystallography where matrices were designated in the form:

(S2 O S1)

Where S1 is input coordinate system, S2 is output coordinate system and O is the operator
corresponding to the matrix. I found this designation is often more useful than the
usual designation by capital letter (which omits information about coordinate systems).

Does anyone know whether this designation is used only in crystallography or is it common in mathematics? Does it have a name? Please provide links if possible.

Hello,

Thank you for your interest in crystallography and the use of matrices in this field. The designation you mentioned, (S2 O S1), is commonly used in crystallography to represent matrices. This notation is also used in mathematics and is known as the "coordinate-free" notation or "Einstein notation". It is a more comprehensive and informative way of representing matrices, as it includes the coordinate systems involved in the transformation.

This notation is not limited to crystallography and is also used in other fields such as physics and engineering. It is especially useful in fields where multiple coordinate systems are involved, as it helps to keep track of the transformations between them.

Here are some links that further explain this notation and its usage:

1. https://en.wikipedia.org/wiki/Einstein_notation
2. https://www.mathworks.com/help/symbolic/einstein-summation.html
3. https://www.sciencedirect.com/topics/chemistry/crystallography

I hope this helps clarify your doubts. Let me know if you have any further questions.

## 1. What is the purpose of designating matrices by (system2 operator system1)?

The purpose of designating matrices by (system2 operator system1) is to represent the transformation of one coordinate system (system1) to another coordinate system (system2) using a matrix multiplication operation. This allows for easier calculation and manipulation of geometric objects and data in different coordinate systems.

## 2. How do you designate a matrix by (system2 operator system1)?

To designate a matrix by (system2 operator system1), you first need to identify the transformation between the two coordinate systems. This can be done by analyzing the relationship between their axes and basis vectors. Then, you can construct a transformation matrix using the transformation coefficients and the appropriate matrix multiplication operation.

## 3. What are the benefits of designating matrices by (system2 operator system1)?

The benefits of designating matrices by (system2 operator system1) include simplifying complex geometric calculations, allowing for easy conversion between coordinate systems, and providing a standardized way to represent transformations. This can be especially useful in fields such as computer graphics, robotics, and physics.

## 4. Can you use any type of matrix for designating by (system2 operator system1)?

Yes, any type of matrix can be used for designating by (system2 operator system1) as long as it represents the appropriate transformation between the two coordinate systems. This can include rotation matrices, translation matrices, and scaling matrices.

## 5. Is it possible to designate matrices by (system2 operator system1) in higher dimensions?

Yes, it is possible to designate matrices by (system2 operator system1) in higher dimensions. The principles and operations used for designating matrices in two or three dimensions can also be applied to higher dimensions, such as four or five dimensions. However, it may become more complex and difficult to visualize the transformations in higher dimensions.

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