“矩阵演算”中的几个基本公式及其应用

Journal of Research of the National Bureau of Standards, Section B: Mathematical Sciences Pub Date : 1971-07-01 DOI:10.6028/JRES.075B.001

J. Fong

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引用次数: 2

摘要

给出了计算单矩阵参数的标量值函数和矩阵值函数梯度的一组初等公式。利用方阵上算子“迹”的一些众所周知的性质，梯度的替代定义和使用乘积法则和微分链式法则计算梯度的简单示例以说明性的方式在分量和矩阵符号中处理，重点是后者。本文以连续介质力学中的两个例子来说明可微函数的“矩阵演算”的应用。/ JR-MS /

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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SOME ELEMENTARY FORMULAS IN "MATRIX CALCULUS" AND THEIR APPLICATIONS

A COLLECTION OF ELEMENTARY FORMULAS FOR CALCULATING THE GRADIENTS OF SCALAR- AND MATRIX-VALUED FUNCTIONS OF ONE MATRIX ARGUMENT IS PRESENTED. USING SOME OF THE WELL-KNOWN PROPERTIES OF THE OPERATOR "TRACE" ON SQUARE MATRICES, ALTERNATIVE DEFINITIONS OF GRADIENTS AND SIMPLE EXAMPLES OF CALCULATING THEM USING THE PRODUCT RULE AND THE CHAIN RULE FOR DIFFERENTIATION ARE TREATED IN AN EXPOSITORY FASHION IN BOTH COMPONENT AND MATRIX NOTATIONS WITH EMPHASIS ON THE LATTER. TWO EXAMPLES IN CONTINUUM MECHANICS ARE PRESENTED TO ILLUSTRATE THE APPLICATON OF THE SO-CALLED "MATRIX CALCULUS" OF DIFFERENTIABLE FUNCTIONS. /JR-MS/

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Journal of Research of the National Bureau of Standards, Section B: Mathematical Sciences

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