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The Matrix Transpose: Visual Intuition

Let's look at what the transpose of a matrix means intuitively. We'll understand how the transpose of a matrix is needed for trying to find pairs of vectors that have the same dot product before and after some linear transformation. We'll also use the Singular Value Decomposition to get a better geometric intuition for how these transformations appear geometrically. #linearalgebra #transpose #svd #SoMEpi Correction: Around 13:20, when I say that Sigma-transpose = Sigma, this is only true if A (and therefore Sigma) are square matrices. Prerequisites: you should already understand how matrices are linear transformations, matrix inverses and the identity matrix, and vector dot products. Knowing about the Singular Value Decomposition would help too, but isn't strictly required. Some good background videos are the Essence of Linear Algebra series by 3Blue1Brown, especially chapters 3 and 9:    • Essence of linear algebra   Chapters: 0:00 Introduction 0:48 Prerequisites 1:19 How to Take the Transpose 1:50 Properties of the Transpose 3:56 Motivating Question 4:56 Linear Transformations Do Not Necessarily Preserve the Dot Product 6:21 Linear Transformations and Dot Products, Visually 7:04 How Can We Preserve the Dot Product? 8:30 Preserved Dot Products, Visually 9:41 Orthogonal Matrices 11:10 Singular Value Decomposition Introduction 12:39 Using the SVD on the Inverse-Transpose 15:28 Additional Examples with the SVD 16:31 What if A is not invertible? 18:25 Main Equation 19:13 Visualization Revisited 19:43 Transpose vs. Inverse 20:38 SVD of the Inverse and Transpose 21:39 SVD of Each Matrix, Visualized 23:34 Symmetric Matrices 24:30 Summary Useful links for learning more: https://en.wikipedia.org/wiki/Transpose    • The deeper meaning of matrix transpose      • Linear Algebra 21j: Two Geometric Int...      • Self-Adjoint Transformations      • Linear Algebra 33 | Transpose and Inn...      • Linear Algebra 59 | Adjoint      • SEE Matrix      • Linear Algebra 60 | Selfadjoint and U...      • Tensor Calculus 12c: The Self-Adjoint...      • Linear Algebra 16c2: A and Aᵀ Have th...      • Linear Operators and their Adjoints   Music by Karl Casey @ White Bat Audio: https://karlcasey.bandcamp.com/ Made with Manim: https://www.manim.community/. The source code can be found at https://github.com/slevey087/transpos... Tips are appreciated! Tip me at: https://ko-fi.com/slevey