Symmetric matrix
As it can be seen from the definition of the matrix, it contains only the values of the elements of a given vector. As it given in (1), we will use zero based indices for vector’s elements and for Tr matrix’s rows and columns: p,q = 0,1,…,n-1. When it is needed to denote from which vector have been obtained the values of Tr matrix’s elements, we will use for the Tr matrix denotation Tr(X), and for its elements - denotation Tr(X) p,q.
Where denotes operation „bitwise exclusive or“ (XOR). Definition of Tr Matrix & Notations Matrix of Transpositions ( Tr matrix) is square matrix with dimension n=2 m, m∈N, which rows are permutations of elements X k, k∈ of given n-dimensional vector X and the values of elements Tr p,q of matrix are obtained from the elements of given vector as follows: Has been investigated the application of Trs matrix for QR decomposition and n-dimensional rotation matrix generation. It has been defined type of matrices of transpositions (Trs matrices) having are mutually orthogonal rows, as Hadamard product of Tr matrix and n-dimensional Hadamard matrix having defined ordering of rows against Sylvester-Hadamard matrix. It has been proved that elements of every two rows makes n/2 fours of elements, in which elements, having different row and column indices (we name them “diagonal elements”) consists the value of the same element of the given vector X. Introduction This article presents a special case of symmetric matrices, matrices of transpositions ( Tr matrices) that are created from the elements of given n-dimensional vector X∈R n, n=2 m, m∈N, where value of every matrix’s element is equal to this element of vector X, which index is equal to bitwise XOR of zero based row and column indices of matrix’s element. Obtaining n-dimensional Hadamard Matrix H(R) n for which the Rows of the Trs Matrix are Mutually Orthogonal 8. The Trs Matrix Created from Unit Vector X is Matrix of Reflection 7. The Trs Matrix has Symmetrical Connectivity of the Elements of the First Row and Column and Antisymmetric Connectivity of the Other Elements Except the Diagonal Elements 6.5. The Trs Matrix Created by a Unit Vector X is an Orthogonal Matrix 6.4. The Fours of Elements in the Hadamard H(R) n-dimensional Matrix must Form Pairs of Orthogonal Two-Dimensional Vectors 6.3.
The Fours of Elements in the Trs Matrix Form Pairs of Orthogonal Two-Dimensional Vectors – Trs-Property 6.2. Fours of Elements in the Trs и H(R) Matrices 6. Matrix of Transpositions with Mutually Orthogonal Rows 5.
Every Two Rows of Tr Matrix Consists n/2 Fours of Elements with the Same Values of the Diagonal Elements 4. First Row & First Column of Tr Matrix Consists Elements of Given Vector X in Normal Order (without Inversions) 3.4. Every One Row and Column of Tr Matrix Consists all n Elements of Given Vector X without Repetition 3.3. (a) The set $S$ consisting of all $n\times n$ symmetric matrices.Article Outline 1. Prove or disprove that the following subsets of $V$ are subspaces of $V$. Let $V$ be the vector space over $\R$ consisting of all $n\times n$ real matrices for some fixed integer $n$. Subspaces of Symmetric, Skew-Symmetric Matrices.The rank of an $m \times n$ matrix $M$ is the dimension of the range (b) If the matrix $B$ is nonsingular, then $\rk(AB)=\rk(A)$. Let $A$ be an $m \times n$ matrix and $B$ be an $n \times l$ matrix. Rank of the Product of Matrices $AB$ is Less than or Equal to the Rank of $A$.The zero vector in $V$ is the $2\times 2$ zero matrix $O$. For any $A\in W$ and $r\in \R$, the scalar product $rA\in W$.For any $A, B\in W$, the sum $A+B\in W$.