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===薄SVD (Thin SVD)===

===薄SVD (Thin SVD)===

矩阵<math>\mathbf{M}</math>的薄SVD(又叫经济型SVD)可表示为:<ref>{{citation | last1=Demmel | first1=James | date=2000|https://www.netlib.org/utk/people/JackDongarra/etemplates/node43.html | title="Decompositions" | last1=Bai | first1=Zhaojun | last2=Dongarra | first2=Jack J. | last3=Ruhe | first3=Axel | last4=van der Vorst | first4=Henk A.|url=https://www.cs.ucdavis.edu/~bai/ET/contents.html| title=Templates for the Solution of Algebraic Eigenvalue Problems | publisher=Society for Industrial and Applied Mathematics | doi=10.1137/1.9780898719581 | isbn=978-0-89871-471-5}}</ref>

矩阵<math>\mathbf{M}</math>的薄SVD(又叫经济型SVD)可表示为:<ref>{{citation

| last1=Demmel

| first1=James

| date=2000

| title="[[Decompositions]]"

| editor1-last=Bai

| editor1-first=Zhaojun

| editor2-first=James

| editor3-last=Dongarra

| editor3-first=Jack J.

| editor4-last=Ruhe

| editor4-first=Axel

| editor5-last=van der Vorst

| editor5-first=Henk A.

| book=[[Templates for the Solution of Algebraic Eigenvalue Problems]]

| publisher=Society for Industrial and Applied Mathematics

| doi=10.1137/1.9780898719581

| isbn=978-0-89871-471-5

}}</ref>

这里<math>k = \min(m,n)</math>，矩阵<math>\mathbf{U}_k</math>和<math>\mathbf{V}_k</math>只包含<math>\mathbf{U}</math>和<math>\mathbf{V}</math>的前k列，<math>\mathbf{\Sigma}_k</math>只包含<math>\mathbf{\Sigma}</math>中的前k个奇异值。因此,矩阵<math>\mathbf{U}_k</math>为<math>m \times k</math>，<math>\mathbf{\Sigma}_k</math>为<math>k \times k</math>对角矩阵，<math>\mathbf{V}_k^*</math>为<math>k \times n</math>。

这里<math>k = \min(m,n)</math>，矩阵<math>\mathbf{U}_k</math>和<math>\mathbf{V}_k</math>只包含<math>\mathbf{U}</math>和<math>\mathbf{V}</math>的前k列，<math>\mathbf{\Sigma}_k</math>只包含<math>\mathbf{\Sigma}</math>中的前k个奇异值。因此,矩阵<math>\mathbf{U}_k</math>为<math>m \times k</math>，<math>\mathbf{\Sigma}_k</math>为<math>k \times k</math>对角矩阵，<math>\mathbf{V}_k^*</math>为<math>k \times n</math>。