- 1、本文档共7页,可阅读全部内容。
- 2、原创力文档(book118)网站文档一经付费(服务费),不意味着购买了该文档的版权,仅供个人/单位学习、研究之用,不得用于商业用途,未经授权,严禁复制、发行、汇编、翻译或者网络传播等,侵权必究。
- 3、本站所有内容均由合作方或网友上传,本站不对文档的完整性、权威性及其观点立场正确性做任何保证或承诺!文档内容仅供研究参考,付费前请自行鉴别。如您付费,意味着您自己接受本站规则且自行承担风险,本站不退款、不进行额外附加服务;查看《如何避免下载的几个坑》。如果您已付费下载过本站文档,您可以点击 这里二次下载。
- 4、如文档侵犯商业秘密、侵犯著作权、侵犯人身权等,请点击“版权申诉”(推荐),也可以打举报电话:400-050-0827(电话支持时间:9:00-18:30)。
查看更多
《million-paper》.pdf
The Hadamard Product
Elizabeth Million
April 12, 2007
1 Introduction and Basic Results
As inexperienced mathematicians we may have once thought that the natural definition for matrix
multiplication would be entrywise multiplication, much in the same way that a young child might
say, “I writed my name.” The mistake is understandable, but it still makes us cringe. Unlike poor
grammar, however, entrywise matrix multiplication has reason to be studied; it has nice properties
in matrix analysis and has applications in both statistics (301 [4], 140 [5]) and physics (93, 149 [5]).
Here we will only expore the properties of the Hadamard product in matrix analysis.
Definition 1.1. Let A and B be m × n matrices with entries in C. The Hadamard product of A
and B is defined by [A ◦ B]ij = [A]ij [B]ij for all 1 ≤ i ≤ m, 1 ≤ j ≤ n .
As we can see, the Hadamard product is simply entrywise multiplication. Because of this, the
Hadamard product inherits the same benefits (and restrictions) of multiplication in C. Note also
that both A and B need to be the same size, but not necessarily square. To avoid confusion,
juxtaposition of matrices will imply the “usual” matrix multiplication, and we will always use “ ◦”
for the Hadamard product.
Now we can explore some basics properties of the Hadamard Product.
Theorem 1.2. Let A and B be m × n matrices with entries in C. Then A ◦ B = B ◦ A.
Proof. The proof follows directly from the fact that multiplication in C is commutative. Let A and
B be m × n matrices with entries in C. Then [A ◦ B]ij = [A]ij [B]ij = [B]ij [A]ij = [B ◦ A]ij and
therefore A ◦ B = B ◦ A.
Theorem 1.3. The identity matrix under the Hadamard product is the m ×n matrix with all entries
equal to 1, denoted J . That is,
您可能关注的文档
- 《MBA产品销售策划学132页doc》.doc
- 《MBA全国联考英语核心单词比较及热门词汇》.doc
- 《MBA十大核心课程》.pdf
- 《MBA历年真题2016-2016》.pdf
- 《MBA培训稿-精益生产》.pdf
- 《MBA市场营销管理》.ppt
- 《MBA教材《商业模式》》.pdf
- 《MBA教程《市场营销学》》.doc
- 《MBA数学常用公式概览(修改后)》.doc
- 《MBA管理沟通集中班——2》.pptx
- 原电池电动势的测定实验报告.pdf
- 与业主、设计、总包、监理和他承包人的配合措施.pdf
- 公司管理流程.pptx
- 2024_2025学年新教材高中地理第1章地球的运动素养综合训练新人教版选择性必修1.doc
- 2024_2025学年新教材高中地理第3章大气的运动第1节常见天气系统第1课时锋与天气分层作业新人教版选择性必修1.doc
- 2024_2025学年新教材高中地理第1章地球的运动第2节地球运动的地理意义第4课时正午太阳高度的变化四季更替和五带划分分层作业课件新人教版选择性必修1.pptx
- 2024_2025学年新教材高中地理第2章地表形态的塑造第2节构造地貌的形成第1课时地质构造与地貌课件新人教版选择性必修1.pptx
- 2024_2025学年新教材高中地理第1章地球的运动问题研究人类是否需要人造月亮课件新人教版选择性必修1.pptx
- 五片小雪花课件.pdf
- 2024_2025学年新教材高中地理第3章大气的运动第2节气压带和风带第1课时气压带和风带的形成分层作业课件新人教版选择性必修1.pptx
文档评论(0)