In mathematics, a unistochastic matrix (also called unitary-stochastic) is a doubly stochastic matrix whose entries are the squares of the absolute values of the entries of some unitary matrix. A square matrix B of size n is doubly stochastic (or bistochastic) if all its entries are non-negative real numbers and each of its rows and columns sum to 1. It is unistochastic if there exists a unitary matrix U such that This definition is analogous to that for an orthostochastic matrix, which is a doubly stochastic matrix whose entries are the squares of the entries in some orthogonal matrix. Since all orthogonal matrices are necessarily unitary matrices, all orthostochastic matrices are also unistochastic. The converse, however, is not true. First, all 2-by-2 doubly stochastic matrices are both unistochastic and orthostochastic, but for larger n this is not the case. For example, take and consider the following doubly stochastic matrix: This matrix is not unistochastic, since any two vectors with moduli equal to the square root of the entries of two columns (or rows) of B cannot be made orthogonal by a suitable choice of phases. For , the set of orthostochastic matrices is a proper subset of the set of unistochastic matrices. * the set of unistochastic matrices contains all permutation matrices and its convex hull is the Birkhoff polytope of all doubly stochastic matrices * for this set is not convex * for the set of triangle inequality on the moduli of the raw is a sufficient and necessary condition for the unistocasticity * for the set of unistochastic matrices is and unistochasticity of any bistochastic matrix B is implied by a non-negative value of its * for the relative volume of the set of unistochastic matrices with respect to the Birkhoff polytope of doubly stochastic matrices is * for explicit conditions for unistochasticity are not known yet, but there exists a numerical method to verify unistochasticity based on the algorithm by Haagerup * The Schur-Horn theorem is equivalent to the following "weak convexity" property of the set of unistochastic matrices: for any vector the set is the convex hull of the set of vectors obtained by all permutations of the entries of the vector (the permutation polytope generated by the vector ). * The set of unistochastic matrices has a nonempty interior. The unistochastic matrix corresponding to the unitary matrix with the entries , where and , is an interior point of . (Wikipedia).
What is a matrix? Free ebook http://tinyurl.com/EngMathYT
From playlist Intro to Matrices
This video introduces the identity matrix and illustrates the properties of the identity matrix. http://mathispower4u.yolasite.com/ http://mathispower4u.wordpress.com/
From playlist Introduction to Matrices and Matrix Operations
How do we add matrices. A matrix is an abstract object that exists in its own right, and in this sense, it is similar to a natural number, or a complex number, or even a polynomial. Each element in a matrix has an address by way of the row in which it is and the column in which it is. Y
From playlist Introducing linear algebra
Identity Matrix | Unit Matrix | Don't Memorise
This video explains the concept of an Identity Matrix. Is it also called a Unit Matrix? ✅To learn more about, Matrices, enroll in our full course now: https://infinitylearn.com/microcourses?utm_source=youtube&utm_medium=Soical&utm_campaign=DM&utm_content=iks8wCfPerU&utm_term=%7Bkeyword%
From playlist Matrices
This video defines a diagonal matrix and then explains how to determine the inverse of a diagonal matrix (if possible) and how to raise a diagonal matrix to a power. Site: mathispower4u.com Blog: mathispower4u.wordpress.com
From playlist Introduction to Matrices and Matrix Operations
We have already looked at the column view of a matrix. In this video lecture I want to expand on this topic to show you that each matrix has a column space. If a matrix is part of a linear system then a linear combination of the columns creates a column space. The vector created by the
From playlist Introducing linear algebra
2 Construction of a Matrix-YouTube sharing.mov
This video shows you how a matrix is constructed from a set of linear equations. It helps you understand where the various elements in a matrix comes from.
From playlist Linear Algebra
Using a set of points determine if the figure is a parallelogram using the midpoint formula
👉 Learn how to determine the figure given four points. A quadrilateral is a polygon with four sides. Some of the types of quadrilaterals are: parallelogram, square, rectangle, rhombus, kite, trapezoid, etc. Each of the types of quadrilateral has its properties. Given four points that repr
From playlist Quadrilaterals on a Coordinate Plane
This video defines elementary matrices and then provides several examples of determining if a given matrix is an elementary matrix. Site: http://mathispower4u.com Blog: http://mathispower4u.wordpress.com
From playlist Augmented Matrices
Lecture 01-03 Linear Algebra review
Machine Learning by Andrew Ng [Coursera] 0113 Matrices and vectors 0114 Addition and scalar multiplication 0115 Matrix-vector multiplication 0116 Matrix-matrix multiplication 0117 Matrix multiplication properties 0118 Inverse and transpose
From playlist Machine Learning by Professor Andrew Ng
The Diagonalization of Matrices
This video explains the process of diagonalization of a matrix.
From playlist The Diagonalization of Matrices
Part IV: Matrix Algebra, Lec 2 | MIT Calculus Revisited: Multivariable Calculus
Part IV: Matrix Algebra, Lecture 2: The "Game" of Matrices Instructor: Herbert Gross View the complete course: http://ocw.mit.edu/RES18-007F11 License: Creative Commons BY-NC-SA More information at http://ocw.mit.edu/terms More courses at http://ocw.mit.edu
From playlist MIT Calculus Revisited: Multivariable Calculus
Desmos Matrix Calc: Matrix Multiplication
This video explains how to us the Desmos Matrix Calculator to perform matrix multiplication. Not solved by hand
From playlist Introduction to Matrices and Matrix Operations
Eigenvectors and Eigenvalues with Jon Krohn
Data scientist Jon Krohn introduces the linear algebra concepts of Eigenvectors and Eigenvalues with a focus on Machine Learning and Python programming. This lesson is an excerpt from “Linear Algebra for Machine Learning LiveLessons” Purchase the entire video course at informit.com/youtub
From playlist Talks and Tutorials
We start discussing how to label matrices and their elements. We then define Order of Matrices and Equal Matrices working an example at 5:49. I then discuss Adding Matrices at 9:40 and work through three examples. Properties of Adding Matrices are explained at 16:00 Scalar Multiplication
From playlist Linear Algebra
Using a Matrix Equation to Solve a System of Equations
This video shows how to solve a system of equations by using a matrix equation. The graphing calculator is integrated into the lesson. http://mathispower4u.yolasite.com/ http://mathispower4u.wordpress.com/
From playlist Matrix Equations
This video defines the transpose of a matrix and explains how to transpose a matrix. The properties of transposed matrices are also discussed. Site: mathispower4u.com Blog: mathispower4u.wordpress.com
From playlist Introduction to Matrices and Matrix Operations
WildLinAlg12: Generalized dilations and eigenvectors
This video introduces the important idea of changing coordinates in Linear Algebra. A linear transformation can be described using many different matrices, depending on the underlying coordinate system, or ordered basis, which is used to describe the space. The simplest case is when the
From playlist A first course in Linear Algebra - N J Wildberger
Differential Equations | Undetermined Coefficients for a System of DEs
We use the method of undetermined coefficients to solve a nonhomogeneous system of first order linear differential equations. http://www.michael-penn.net http://www.randolphcollege.edu/mathematics/
From playlist Systems of Differential Equations
Singular Value Decomposition (SVD) and Image Compression
Github repo: http://www.github.com/luisguiserrano/singular_value_decomposition Grokking Machine Learning Book: https://www.manning.com/books/grokking-machine-learning 40% discount promo code: serranoyt In this video, we learn a very useful matrix trick called singular value decomposition
From playlist Unsupervised Learning