向量 矩陣 張量

機器學習代數 (MACHINE LEARNING ALGEBRA)

Algebra is an important element of mathematics and has a lot of practical applications. Among other things, it plays a crucial role in the economy, quantum computing, and machine learning. For the latter one, matrices and vectors are important, while the popular Python framework PyTorch uses tensor-based operations. Despite their similarities, a tensor is neither a matrix nor a vector, contrary to what many people think.

代數是數學的重要組成部分，具有許多實際應用。 除其他外，它在經濟，量子計算和機器學習中起著至關重要的作用。 對于后者，矩陣和向量很重要，而流行的Python框架PyTorch使用基于張量的運算。 盡管有相似之處，但張量既不是矩陣也不是矢量，這與許多人的想法相反。

A matrix is a grid of m x n numbers surrounded by square brackets. Here, m is the number of rows and n is the number of columns. Mathematical operations can be performed on matrices, such as e.g. matrix multiplication, matrix addition, and many more.

矩陣是由方括號包圍的mxn數字網格。 此處，m是行數，n是列數。 可以在矩陣上執行數學運算，例如矩陣乘法，矩陣加法等等。

A vector is a 1D array of numbers, a matrix where m or n is equal to 1. Similarly to a matrix, it is also possible to perform numerous mathematical operations on a vector, and it is possible to multiply matrices with vectors and vice versa.

向量是一維數字數組，其中m或n等于1的矩陣。類似于矩陣，還可以對向量執行大量數學運算，并且可以將矩陣與向量相乘，反之亦然。

A tensor, however, can be thought of as a generalized matrix which can be described by its rank. The rank of a tensor is an integer number of 0 or higher. A tensor with rank 0 can be represented by a scalar, a tensor with rank 1 can be represented by a vector and a tensor of rank 2 can be represented by a matrix. There are also tensors of rank 3 and higher, the latter ones being more difficult to visualize. In addition to the rank, there are certain characteristics of tensors related to how they interact with other mathematical entities. If one of the entities in an interaction transform the other entity or entities, then the tensor has to obey a related transformation rule.

但是，張量可以認為是可以用其秩來描述的廣義矩陣。 張量的秩是0或更高的整數。 等級為0的張量可以由標量表示，等級為1的張量可以由矢量表示，等級2的張量可以由矩陣表示。 還存在3級或更高的張量，后者更難以可視化。 除等級外，張量還具有某些與張量與其他數學實體的相互作用有關的特征。 如果交互中的一個實體變換了另一個實體，則張量必須服從相關的變換規則。

[1] Steven Steinke. What’s the difference between a matrix and a tensor? (Aug 2017). https://medium.com/@quantumsteinke/whats-the-difference-between-a-matrix-and-a-tensor-4505fbdc576c

[1]史蒂文·斯坦克。 矩陣和張量之間有什么區別？ (2017年8月)。 https://medium.com/@quantumsteinke/whats-the-difference-between-a-matrix-and-a-tensor-4505fbdc576c

翻譯自: https://medium.com/swlh/what-is-the-difference-between-a-tensor-a-matrix-and-a-vector-ce9982f35064

向量 矩陣 張量

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