All have a singular value decomposition, meaning there exists an orthonormal basis and such that, where the's are

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All<br><img src='https://www.fatskills.com/images2/GradExams/9EDFE78D-3C4D-4DE6-B431-1FF9AC8560BF.png' height='20' width='81'/> have a singular value decomposition, meaning there exists an orthonormal basis<br><img src='https://www.fatskills.com/images2/GradExams/83C03782-637F-469A-A7FD-E3D9986846E4.png' height='20' width='153'/> and<br><img src='https://www.fatskills.com/images2/GradExams/8E2DE55E-249A-4680-9889-B07B377B3070.png' height='20' width='157'/> such that<br><img src='https://www.fatskills.com/images2/GradExams/48039D04-797B-4F70-8D51-ABA2C888D7C1.png' height='20' width='313'/>, where the<br><img src='https://www.fatskills.com/images2/GradExams/91BB2A69-740B-42AB-9954-69BFF0EB2DF4.png' height='12' width='13'/>'s are