On the existence of equiangular tight frames
... (1) (Orthonormal Bases). When N = d, the sole examples of ETFs are unitary (and orthogonal) matrices. Evidently, the absolute inner product α between distinct vectors is zero. (2) (Simplices). When N = d + 1, every ETF can be viewed as the vertices of a regular simplex centered at the origin [7,17]. ...
... (1) (Orthonormal Bases). When N = d, the sole examples of ETFs are unitary (and orthogonal) matrices. Evidently, the absolute inner product α between distinct vectors is zero. (2) (Simplices). When N = d + 1, every ETF can be viewed as the vertices of a regular simplex centered at the origin [7,17]. ...
Matrix Lie groups and their Lie algebras
... (a) The general linear group GL(n): This is clearly a matrix Lie group. (b) The special linear group SL(n): To see this is a matrix Lie group note that if {Ak } is a sequence in SL(n), det(Ak ) = 1, such that Ak → A, then by continuity of the determinant det(A) = 1 also; therefore, A ∈ SL(n). (c) Th ...
... (a) The general linear group GL(n): This is clearly a matrix Lie group. (b) The special linear group SL(n): To see this is a matrix Lie group note that if {Ak } is a sequence in SL(n), det(Ak ) = 1, such that Ak → A, then by continuity of the determinant det(A) = 1 also; therefore, A ∈ SL(n). (c) Th ...
Rotation matrix
... the quadratic), and whose sum is 2 cos θ (the negated linear term). This factorization is of interest for 3×3 rotation matrices because the same thing occurs for all of them. (As special cases, for a null rotation the "complex conjugates" are both 1, and for a 180° rotation they are both −1.) Furthe ...
... the quadratic), and whose sum is 2 cos θ (the negated linear term). This factorization is of interest for 3×3 rotation matrices because the same thing occurs for all of them. (As special cases, for a null rotation the "complex conjugates" are both 1, and for a 180° rotation they are both −1.) Furthe ...
Mathematical Foundations for Computer Science I B.sc., IT
... diagonal matrix all the entries except the entries along the main diagonal is zero. 2) Triangular matrix A square matrix in which all the entries above the main diagonal are zero is called a lower triangular matrix. If all the entries below the main diagonal are zero, it is called an upper triangula ...
... diagonal matrix all the entries except the entries along the main diagonal is zero. 2) Triangular matrix A square matrix in which all the entries above the main diagonal are zero is called a lower triangular matrix. If all the entries below the main diagonal are zero, it is called an upper triangula ...
Vector Space
... vector ez points in the z direction. There are several common notations for these vectors, including {ex, ey, ez}, {e1, e2, e3}, {i, j, k}, and {x, y, z}. In addition, these vectors are sometimes written with a hat to emphasize their status as unit vectors. These vectors are a basis in the sense tha ...
... vector ez points in the z direction. There are several common notations for these vectors, including {ex, ey, ez}, {e1, e2, e3}, {i, j, k}, and {x, y, z}. In addition, these vectors are sometimes written with a hat to emphasize their status as unit vectors. These vectors are a basis in the sense tha ...
Lecture6
... 1. Construct an augmented matrix for the given system of equations. 2. Use elementary row operations to transform the augmented matrix into an augmented matrix in row-reduced form. 3. Write the equations associated with the resulting augmented matrix. 4. Solve the new set of equations by bac ...
... 1. Construct an augmented matrix for the given system of equations. 2. Use elementary row operations to transform the augmented matrix into an augmented matrix in row-reduced form. 3. Write the equations associated with the resulting augmented matrix. 4. Solve the new set of equations by bac ...
Implementing Sparse Matrices for Graph Algorithms
... may not hold. In this chapter, however, we use this assumption in our analysis. In contrast, Buluç and Gilbert gave an SpGEMM algorithm specifically designed for ...
... may not hold. In this chapter, however, we use this assumption in our analysis. In contrast, Buluç and Gilbert gave an SpGEMM algorithm specifically designed for ...
lecture
... If two sequences have a different last character, the length of the LCS is either the length of the LCS we get by dropping the last character from the first sequence, or the last character from the second sequence ...
... If two sequences have a different last character, the length of the LCS is either the length of the LCS we get by dropping the last character from the first sequence, or the last character from the second sequence ...
