In linear algebra, an orthogonal matrix, or orthonormal matrix, is a real square matrix whose columns and rows are orthonormal vectors. One way to express this is where QT is the transpose of Q and I is the identity matrix. This leads to the equivalent characterization: a matrix Q is orthogonal if its transpose is equal to its inverse: where Q−1 is the inverse of Q. An orthogonal matrix Q is necessarily invertible (with inverse Q−1 = QT), unitary (Q−1 = Q∗), where Q∗ is the Hermitian adjoint (conjugate transpose) of Q, and therefore normal (Q∗Q = QQ∗) over the real numbers. The determinant of any orthogonal matrix is either +1 or −1. As a linear transformation, an orthogonal matrix preserves the inner product of vectors, and therefore acts as an isometry of Euclidean space, such as a rotation, reflection or rotoreflection. In other words, it is a unitary transformation. The set of n × n orthogonal matrices, under multiplication, forms the group O(n), known as the orthogonal group. The subgroup SO(n) consisting of orthogonal matrices with determinant +1 is called the special orthogonal group, and each of its elements is a special orthogonal matrix. As a linear transformation, every special orthogonal matrix acts as a rotation. (Wikipedia).
Linear Algebra 7.1 Orthogonal Matrices
My notes are available at http://asherbroberts.com/ (so you can write along with me). Elementary Linear Algebra: Applications Version 12th Edition by Howard Anton, Chris Rorres, and Anton Kaul A. Roberts is supported in part by the grants NSF CAREER 1653602 and NSF DMS 2153803.
From playlist Linear Algebra
11H Orthogonal Projection of a Vector
The orthogonal projection of one vector along another.
From playlist Linear Algebra
Orthogonal Matrix Example (Ch5 Pr28)
We look at a rotation matrix as an example of a orthogonal matrix. This is Chapter 5 Problem 28 from the MATH1141/MATH1131 Algebra notes. Presented by Daniel Mansfield from the School of Mathematics and Statistics at UNSW.
From playlist Mathematics 1A (Algebra)
Orthogonal matrices | Lecture 7 | Matrix Algebra for Engineers
Definition of orthogonal matrices. Join me on Coursera: https://www.coursera.org/learn/matrix-algebra-engineers Lecture notes at http://www.math.ust.hk/~machas/matrix-algebra-for-engineers.pdf Subscribe to my channel: http://www.youtube.com/user/jchasnov?sub_confirmation=1
From playlist Matrix Algebra for Engineers
11J Orthogonal Projection of a Vector
The orthogonal projection of one vector along another.
From playlist Linear Algebra
Linear Algebra 21j: Two Geometric Interpretations of Orthogonal Matrices
https://bit.ly/PavelPatreon https://lem.ma/LA - Linear Algebra on Lemma http://bit.ly/ITCYTNew - Dr. Grinfeld's Tensor Calculus textbook https://lem.ma/prep - Complete SAT Math Prep
From playlist Part 3 Linear Algebra: Linear Transformations
Proof: Orthogonal Matrices Satisfy A^TA=I
One way to characterize orthogonal matrices is to say that a matrix orthogonal if and only if A transpose times A is the identity matrix. In this video, we prove this result using basic matrix calculations and the definition of orthonormal vectors. Learning Linear Algebra playlist: https:
From playlist Learning Linear Algebra
11I Orthogonal Projection of a Vector
The Orthogonal Projection of one vector along another.
From playlist Linear Algebra
Principal axes theorem + orthogonal matrices
Free ebook http://tinyurl.com/EngMathYT A basic introduction to orthogonal matrices and the principal axes theorem. Several examples are presented involving a simplification of quadratic (quadric) forms. A proof is also given.
From playlist Engineering Mathematics
Linear Algebra 20j: The Dot Product, Matrix Multiplication, and the Magic of Orthogonal Matrices
https://bit.ly/PavelPatreon https://lem.ma/LA - Linear Algebra on Lemma http://bit.ly/ITCYTNew - Dr. Grinfeld's Tensor Calculus textbook https://lem.ma/prep - Complete SAT Math Prep
From playlist Part 3 Linear Algebra: Linear Transformations
3. Orthonormal Columns in Q Give Q'Q = I
MIT 18.065 Matrix Methods in Data Analysis, Signal Processing, and Machine Learning, Spring 2018 Instructor: Gilbert Strang View the complete course: https://ocw.mit.edu/18-065S18 YouTube Playlist: https://www.youtube.com/playlist?list=PLUl4u3cNGP63oMNUHXqIUcrkS2PivhN3k This lecture focus
From playlist MIT 18.065 Matrix Methods in Data Analysis, Signal Processing, and Machine Learning, Spring 2018
MIT 18.06 Linear Algebra, Spring 2005 Instructor: Gilbert Strang View the complete course: http://ocw.mit.edu/18-06S05 YouTube Playlist: https://www.youtube.com/playlist?list=PLE7DDD91010BC51F8 32. Quiz 3 Review License: Creative Commons BY-NC-SA More information at https://ocw.mit.edu/t
From playlist MIT 18.06 Linear Algebra, Spring 2005
17. Orthogonal Matrices and Gram-Schmidt
MIT 18.06 Linear Algebra, Spring 2005 Instructor: Gilbert Strang View the complete course: http://ocw.mit.edu/18-06S05 YouTube Playlist: https://www.youtube.com/playlist?list=PLE7DDD91010BC51F8 17. Orthogonal Matrices and Gram-Schmidt License: Creative Commons BY-NC-SA More information a
From playlist MIT 18.06 Linear Algebra, Spring 2005
Lec 13 | MIT 18.085 Computational Science and Engineering I
Numerical linear algebra: orthogonalization and A = QR A more recent version of this course is available at: http://ocw.mit.edu/18-085f08 License: Creative Commons BY-NC-SA More information at http://ocw.mit.edu/terms More courses at http://ocw.mit.edu
From playlist MIT 18.085 Computational Science & Engineering I, Fall 2007
36 entangled officers of Euler: A quantum solution to a classically... by Arul Lakshminarayan
Colloquium: 36 entangled officers of Euler: A quantum solution to a classically impossible problem Speaker: Arul Lakshminarayan (IIT Madras, Chennai) Date: Mon, 06 June 2022, 15:30 to 17:00 Venue: Online and Madhava Lecture Hall Abstract The 36 officers problem of Euler is a well-known i
From playlist ICTS Colloquia
P2L04_15Oct2020 (Tut6, Q6,11) (un-edited)
P2L04_15Oct2020: 4th lecture, zoom (un-edited), Tut 6, Q6 & 11 (recorded 15 Oct 2020) Quest 6) 23a 00:00 - 02:25 23b 02:25 - 08:15 23c 08:39 - 13:05 23d 13:05 - 16:30 23e 16:40 - 24:30 25:03 - 25:26 "How do we justify 23d" 24a 27:00 - 28:05 24b 28
From playlist Part 2 lectures (2020 zoom)
MATH2018 Lecture 6.2 Special Matrices
We look at the properties of invertible matrices, symmetric matrices, and orthogonal matrices, and discuss some important relationships between them.
From playlist MATH2018 Engineering Mathematics 2D
Linear Algebra - Lecture 41 - Diagonalization of Symmetric Matrices
In this lecture, we investigate the diagonalization of symmetric matrices.
From playlist Linear Algebra Lectures
31. Eigenvectors of Circulant Matrices: Fourier Matrix
MIT 18.065 Matrix Methods in Data Analysis, Signal Processing, and Machine Learning, Spring 2018 Instructor: Gilbert Strang View the complete course: https://ocw.mit.edu/18-065S18 YouTube Playlist: https://www.youtube.com/playlist?list=PLUl4u3cNGP63oMNUHXqIUcrkS2PivhN3k This lecture conti
From playlist MIT 18.065 Matrix Methods in Data Analysis, Signal Processing, and Machine Learning, Spring 2018
The Diagonalization of Matrices
This video explains the process of diagonalization of a matrix.
From playlist The Diagonalization of Matrices