We investigated some difficulties that students often face when studying
linear algebra at the undergraduate level, and identified some common mistakes
and difficulties they often encountered when dealing with topics that require
algorithmic thinking skills such as matrix factorization. In particular, we
focused on (orthogonal) diagonalization and singular value decomposition (SVD).
We also offered the possibility of exploring these topics using SageMath, a
Python-based free open software computer algebra system (CAS) that has been
identified to be useful for assisting many students in the computational
process even though its output is static by nature. We then explored dynamic
ChatGPT by inquiring the chatbot about the topic, either by asking to provide
an example or to solve a problem, that is by constructing an (orthogonal)
diagonalization or SVD from a particular matrix. By consolidating essential
concepts in linear algebra and improving computational skills through effective
practice, mastering these topics would become easier and mistakes could be
minimized. Static SageMath, in particular, is a great aid for calculation
confirmation and handling tedious computations. Although dynamic ChatGPT is
relatively unreliable for solving problems in linear algebra, the mistakes it
produces could become a valuable tool for improving critical thinking skills.

研究生在学习线性代数时通常会遇到的难点及解决方式，包括使用免费计算机代数系统 SageMath 处理需要算法思维技能的问题，如对角化和奇异值分解，并通过 ChatGPT 提高计算能力和批判性思维。

矩阵对角化和奇异值分解：静态 SageMath 和动态 ChatGPT 并置

Matrix diagonalization and singular value decomposition: Static SageMath and dynamic ChatGPT juxtaposed

Diagonalization, or eigenvalue decomposition, is very useful in many areas of
applied mathematics, including signal processing and quantum physics. Matrix
decomposition is also a useful tool for approximating matrices as the product
of a matrix and its transpose, which relates to unitary diagonalization. As
stated by the spectral theorem, only normal matrices are unitarily
diagonalizable. However we show that all real square matrices are the real part
of some unitarily diagonalizable matrix.

本文介绍了矩阵对角化中的一些重要概念，探讨了只有正交对角矩阵具有谱分解的证明，并证明了所有实方阵都可通过正交相似化为实数部分相同的正交矩阵。

通过幺正对角化分解实方阵

Decomposing Real Square Matrices via Unitary Diagonalization

Algebraic geometry, although little explored in signal processing, provides
tools that are very convenient for investigating generic properties in a wide
range of applications. Generic properties are properties that hold "almost
everywhere". We present a set of conditions that are sufficient for
demonstrating the generic uniqueness of a certain structured matrix
factorization. This set of conditions may be used as a checklist for generic
uniqueness in different settings. We discuss two particular applications in
detail. We provide a relaxed generic uniqueness condition for joint matrix
diagonalization that is relevant for independent component analysis in the
underdetermined case. We present generic uniqueness conditions for a recently
proposed class of deterministic blind source separation methods that rely on
mild source models. For the interested reader we provide some intuition on how
the results are connected to their algebraic geometric roots.

应用代数几何中提供的工具可以方便研究各种常见通用问题，本文介绍了一组条件用于证明特定结构矩阵分解的通用唯一性，并提供了两种详细的应用场景：基于矩阵对角化的独立成分分析中的通用唯一性条件和基于轻微源模型的盲源分离方法的通用唯一性条件。