orthogonalize

简明释义

英[ɔrˈθɒɡənəˌlaɪz]美[ɔrˈθɒɡənəˌlaɪz]

vt. 使成为正交；使正交化

第 三 人 称 单 数 o r t h o g o n a l i z e s

现 在 分 词 o r t h o g o n a l i z i n g

过 去 式 o r t h o g o n a l i z e d

过 去 分 词 o r t h o g o n a l i z e d

英英释义

单词用法

同义词

反义词

例句

1.In linear algebra, we often need to orthogonalize 正交化 a set of vectors to simplify calculations.

在线性代数中，我们经常需要对一组向量进行orthogonalize 正交化以简化计算。

2.The process to orthogonalize 正交化 data helps in reducing redundancy in machine learning models.

对数据进行orthogonalize 正交化的过程有助于减少机器学习模型中的冗余。

3.Before performing regression analysis, it's crucial to orthogonalize 正交化 the predictor variables.

在进行回归分析之前，关键是要对预测变量进行orthogonalize 正交化。

4.To improve the stability of numerical methods, we can orthogonalize 正交化 the basis functions used.

为了提高数值方法的稳定性，我们可以对所用的基函数进行orthogonalize 正交化。

5.The Gram-Schmidt process is a common technique used to orthogonalize 正交化 a set of linearly independent vectors.

Gram-Schmidt过程是一种常用的技术，用于对一组线性无关的向量进行orthogonalize 正交化。

作文

In the field of mathematics and computer science, the concept of orthogonality plays a pivotal role in various applications. To understand this concept better, we often need to employ techniques that help us to orthogonalize vectors or functions. The term orthogonalize refers to the process of transforming a set of vectors into a new set of vectors that are mutually perpendicular, or orthogonal, to one another. This is crucial in many areas such as linear algebra, signal processing, and machine learning. For instance, consider a scenario in linear algebra where we have a set of linearly independent vectors. These vectors can be represented in a multi-dimensional space. However, they may not necessarily be orthogonal. By applying the Gram-Schmidt process, we can orthogonalize these vectors. The Gram-Schmidt process takes each vector in the original set and subtracts the projections of the previously orthogonalized vectors from it. The result is a new set of vectors that are orthogonal to each other, simplifying many calculations and ensuring numerical stability in computations. In the realm of signal processing, orthogonalize also has significant implications. When dealing with signals, we often want to ensure that different signals do not interfere with one another. By orthogonalizing the signals, we can achieve clearer communication channels. For example, in Orthogonal Frequency Division Multiplexing (OFDM), multiple signals are transmitted simultaneously without interference by ensuring they are orthogonal to each other. This allows for more efficient use of bandwidth and improves signal integrity. Machine learning also benefits from the concept of orthogonalize. In high-dimensional datasets, features may be correlated, leading to redundancy in information. By orthogonalizing the features, we can reduce this redundancy, improve model performance, and enhance interpretability. Techniques such as Principal Component Analysis (PCA) effectively orthogonalize features, allowing us to capture the most significant patterns in the data while discarding noise. Moreover, the process of orthogonalize is not limited to mathematical or technical applications. It can also be applied metaphorically in various aspects of life. For example, in teamwork, individuals often bring diverse perspectives and skills to the table. By finding ways to orthogonalize these contributions, teams can work more effectively together. Each member's unique strengths can complement one another, leading to innovative solutions and a harmonious working environment. In conclusion, the ability to orthogonalize is an essential skill across various disciplines. Whether in mathematics, engineering, or everyday teamwork, understanding how to create orthogonal relationships can lead to enhanced clarity, efficiency, and effectiveness. As we continue to explore and apply the concept of orthogonalize, we open doors to new possibilities and innovations that can significantly impact our world. Thus, mastering the idea of orthogonalize is not only beneficial but also necessary in today's increasingly complex landscape.

在数学和计算机科学领域，正交性这一概念在各种应用中扮演着重要角色。为了更好地理解这一概念，我们常常需要采用一些技术来帮助我们正交化向量或函数。术语正交化是指将一组向量转换为另一组彼此垂直（或正交）的新向量的过程。这在许多领域中都是至关重要的，例如线性代数、信号处理和机器学习。 例如，考虑在线性代数中，我们有一组线性无关的向量。这些向量可以在多维空间中表示。然而，它们可能并不一定是正交的。通过应用Gram-Schmidt过程，我们可以正交化这些向量。Gram-Schmidt过程取原始集合中的每个向量，并从中减去之前正交化向量的投影。结果是一个新的向量集合，这些向量彼此正交，从而简化了许多计算，并确保计算的数值稳定性。 在信号处理领域，正交化也具有重要意义。在处理信号时，我们通常希望确保不同的信号不会相互干扰。通过正交化信号，我们可以实现更清晰的通信通道。例如，在正交频分复用（OFDM）中，同时传输多个信号而不发生干扰，通过确保它们彼此正交。这允许更有效地使用带宽，并提高信号完整性。 机器学习也受益于正交化的概念。在高维数据集中，特征可能存在相关性，导致信息冗余。通过正交化特征，我们可以减少这种冗余，提高模型性能，并增强可解释性。主成分分析（PCA）等技术有效地正交化特征，使我们能够捕捉数据中最显著的模式，同时丢弃噪声。 此外，正交化的过程不仅限于数学或技术应用。在生活的各个方面，它也可以隐喻性地应用。例如，在团队合作中，个人往往会带来多样的视角和技能。通过找到方法来正交化这些贡献，团队可以更有效地协作。每个成员的独特优势可以相互补充，导致创新解决方案和和谐的工作环境。 总之，正交化的能力是各学科中一项基本技能。无论是在数学、工程还是日常团队合作中，理解如何创建正交关系都可以带来更高的清晰度、效率和效果。随着我们继续探索和应用正交化的概念，我们为新的可能性和创新打开了大门，这些可能性和创新可以显著影响我们的世界。因此，掌握正交化的思想不仅是有益的，而且在当今日益复杂的环境中也是必要的。