Tag: MathNov 16, 2021酉矩阵为什么叫酉矩阵 酉矩阵为什么叫酉矩阵 酉矩阵为什么叫酉矩阵 2021-11-16 Tags: #Math/LinearAlgebra #English 酉矩阵里面的"酉"其实是字母UUU的音译 （又译作幺正矩阵，英语：unitary matrix） unitary adjective uk /ˈjuː.nɪ.tər.i/ us /ˈjuː.nɪ.ter.i/ of a system of local government in the UK in which official... 11/19/2023 AllMath/LinearAlgebraEnglishNov 14, 2021MIT_18.065-Part_7-Eigenvalues and Eigenvectors MIT_18.065-Part_7-Eigenvalues and Eigenvectors Eigenvalues and Eigenvectors 2021-11-14 Tags: #Math/LinearAlgebra #Math/LinearAlgebra/Eigenvalue 特征值的一些性质 相似矩阵 相似矩阵有相同的特征值 P−1AP=BP^{-1}AP = BP−1AP=B 假设矩阵BBB有特征值λ\lambdaλ: Bx=λxBx=\lambda xBx=λx 则 P−1APx=λx⇒APx=Pλx⇒A(Px)=λ(Px)\begin{aligned}P^{-1}APx&=\lambda x \\ &\Rightarrow \\ APx&=P\lambda x \\ &\Rightarrow \\ A(Px)&=\lambda (Px)\end{aligned}P−1APxAPxA(Px)=λx⇒=Pλx⇒=λ(Px) A也有特征值λ\lambdaλ,... 11/19/2023 AllMath/LinearAlgebraMath/LinearAlgebra/EigenvalueNov 14, 2021MIT_18.065-Part_7-Eigenvalues and Eigenvectors MIT_18.065-Part_7-Eigenvalues and Eigenvectors Eigenvalues and Eigenvectors 2021-11-14 Tags: #Math/LinearAlgebra #Math/LinearAlgebra/Eigenvalue 特征值的一些性质 相似矩阵 相似矩阵有相同的特征值 P−1AP=BP^{-1}AP = BP−1AP=B 假设矩阵BBB有特征值λ\lambdaλ: Bx=λxBx=\lambda xBx=λx 则 P−1APx=λx⇒APx=Pλx⇒A(Px)=λ(Px)\begin{aligned}P^{-1}APx&=\lambda x \\ &\Rightarrow \\ APx&=P\lambda x \\ &\Rightarrow \\ A(Px)&=\lambda (Px)\end{aligned}P−1APxAPxA(Px)=λx⇒=Pλx⇒=λ(Px) A也有特征值λ\lambdaλ,... 11/19/2023 AllMath/LinearAlgebraMath/LinearAlgebra/EigenvalueNov 14, 2021MIT_18.065-Part_8-Positive Definite and Semidefinite Matrices MIT_18.065-Part_8-Positive Definite and Semidefinite Matrices Positive Definite and Semi-definite Matrices 2021-11-14 Tags: #Math/LinearAlgebra Positive Definite Matrices are the Best of the Symmetric Matrices. 五个判别条件 同时也是正定矩阵的重要性质: Positive Eigenvalues 所有的特征值都是正数 Energy xTSx>0x^TSx>0xTSx>0, $\forall... 11/19/2023 AllMath/LinearAlgebraNov 14, 2021MIT_18.065-Part_9-Singular Value Decomposition-SVD MIT_18.065-Part_9-Singular Value Decomposition-SVD Singular Value Decomposition 2021-11-14 Tags: #Math/LinearAlgebra #SVD SVD将任意矩阵AAA分解成了三个部分: UUU 的列是相互正交的. - Left Singular Vectors Σ\SigmaΣ 的对角线上面是递减的奇异值. - Singular Values VTV^TVT 的行是相互正交的. - Right... 11/19/2023 AllMath/LinearAlgebraSVDNov 14, 2021理解相似矩阵 理解相似矩阵 理解相似矩阵 2021-11-14 Tags: #Math/LinearAlgebra 设 A,BA,BA,B 都是 nnn 阶矩阵，若有可逆矩阵 PPP , 使得 B=P−1APB=P^{-1}APB=P−1AP , 则称BBB是AAA的相似矩阵。 相似矩阵是同一个线性变换在不同基向量下的不同矩阵表示. PPP是基变换矩阵(Base Change Matrix), 它是一个不改变空间维数的可逆线性变换, 其目的是改变当前线性空间的基底: $(\space \vec... 11/19/2023 AllMath/LinearAlgebraNov 13, 2021MIT_18.065-Part_5-Four_Subspaces MIT_18.065-Part_5-Four_Subspaces 4 Subspaces 2021-11-13 Tags: #Math/LinearAlgebra What are they? The Four Subspaces are: AAA的列空间 Col(A)Col(A)Col(A) AAA的行空间 Row(A)=Col(AT)Row(A)=Col(A^T)Row(A)=Col(AT) AAA的核(零空间) N(A)⇒Ax=0N(A) \Rightarrow Ax=0N(A)⇒Ax=0 ATA^TAT的核 N(AT)⇒ATy=0N(A^T) \Rightarrow A^Ty=0N(AT)⇒ATy=0 为了与课本一致, 我们将Col(A)Col(A)Col(A)简称为C(A)C(A)C(A) What do... 11/19/2023 AllMath/LinearAlgebraNov 13, 2021MIT_18.065-Part_6-Orthonormal Columns in Q Give Q'Q = I MIT_18.065-Part_6-Orthonormal Columns in Q Give Q'Q = I Orthonormal Columns in Q Give Q'Q = I 2021-11-13 Tags: #Math/LinearAlgebra orthogonal: 正交的 orthonormal: Ortho(gonal) + normal, 即又正交又是单位向量, 长度为1. Orthogonal Vectors 内积的向量化表示: $\boldsymbol{x}^{\mathrm{T}}... 11/19/2023 AllMath/LinearAlgebraNov 12, 2021MIT 18.065 Matrix Methods in Data Analysis, Signal Processing, and Machine Learning, MIT 18.065 Matrix Methods in Data Analysis, Signal Processing, and Machine Learning, MIT 18.065 - Matrix Methods in Data Analysis, Signal Processing, and Machine Learning, 2021-11-12 Tags: #Matrix #Math/LinearAlgebra #Math 学习这门课的主要动力是线性代数的知识在大二的一年内已经有所遗忘了, 并且在学习机器学习期间常常设计线性代数与矩阵的相关知识, 所以想要有针对性地深入学习与复习一下. 也想体验一下Gilbert... 11/19/2023 AllMatrixMath/LinearAlgebraMathNov 12, 2021MIT 18.065 Matrix Methods in Data Analysis, Signal Processing, and Machine Learning, MIT 18.065 Matrix Methods in Data Analysis, Signal Processing, and Machine Learning, MIT 18.065 - Matrix Methods in Data Analysis, Signal Processing, and Machine Learning, 2021-11-12 Tags: #Matrix #Math/LinearAlgebra #Math 学习这门课的主要动力是线性代数的知识在大二的一年内已经有所遗忘了, 并且在学习机器学习期间常常设计线性代数与矩阵的相关知识, 所以想要有针对性地深入学习与复习一下. 也想体验一下Gilbert... 11/19/2023 AllMatrixMath/LinearAlgebraMath«««23456»»»