Reedeegi said:How would you represent tensors as matrices? I've searched all over, and my book on GR (Wald) only has one example where he makes a matrice from a tensor, and I still don't understand the process.
George Jones said:A tensor is a muti-linear mappring. A rank 2 tensor can be turned into a matrix by letting it operate on a basis or dual basis. See,
https://www.physicsforums.com/showthread.php?p=874061&#post874061
but this might not have enough detail.
Reedeegi said:how often are rank-2 tensors used in GR, by chance?
George Jones said:Quite often; the metric tensor, the stress-energy tensor, the Einstein tensor, and the Ricci tensor all examples of rank 2 tensors.
Introducing a basis turns (the components of):
a rank 1 tensor into a column or row of values;
a rank 2 matrix into a square matrix of values;
a rank 3 tensor into a cube with values in each subcubes (think rubik's cube).
A tensor is a mathematical object that describes the linear relationship between different sets of data. It is important in science because it allows us to represent complex physical quantities and relationships in a concise and efficient manner. Tensors are used in many areas of science, including physics, engineering, and computer science.
A tensor can be represented in matrix form by creating a multidimensional array of numbers that correspond to the different components of the tensor. The number of dimensions in the array depends on the rank of the tensor, which is determined by the number of indices needed to fully describe its components.
Matrix representation of tensors can be useful in data analysis because it allows us to perform operations such as multiplication, addition, and inversion on tensors, which can help us to extract and analyze important information from large datasets. Additionally, matrix representation of tensors can be easily visualized and manipulated using computer software, making it a powerful tool for data analysis.
A tensor is a more general mathematical object than a matrix. While a matrix has two dimensions (rows and columns), a tensor can have any number of dimensions. Additionally, each element in a tensor can be a vector or another tensor, whereas in a matrix, each element is a single number.
Yes, tensors can be represented in other forms besides matrices. They can also be represented as arrays, graphs, or in index notation. However, the matrix representation is the most commonly used and most convenient form for performing mathematical operations on tensors.