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For a square matrix, the diagonal (or main diagonal or principal diagonal) is the diagonal line of entries running from the top-left corner to the bottom-right corner. [ 1 ] [ 2 ] [ 3 ] For a matrix A {\displaystyle A} with row index specified by i {\displaystyle i} and column index specified by j {\displaystyle j} , these would be entries A i ...
The diagonals of a cube with side length 1. AC' (shown in blue) is a space diagonal with length , while AC (shown in red) is a face diagonal and has length .. In geometry, a diagonal is a line segment joining two vertices of a polygon or polyhedron, when those vertices are not on the same edge.
A magic square is an arrangement of numbers in a square grid so that the sum of the numbers along every row, column, and diagonal is the same. Similarly, one may define a magic cube to be an arrangement of numbers in a cubical grid so that the sum of the numbers on the four space diagonals must be the same as the sum of the numbers in each row, each column, and each pillar.
One way to visualize a broken diagonal is to imagine a "ghost image" of the panmagic square adjacent to the original: The set of numbers {3, 12, 14, 5} of a broken diagonal, wrapped around the original square, can be seen starting with the first square of the ghost image and moving down to the left.
is the unit circle in P 2 = is the unit hyperbola in P 2. + + + = gives the Fermat cubic surface in P 3 with 27 lines. The 27 lines in this example are easy to describe explicitly: they are the 9 lines of the form (x : ax : y : by) where a and b are fixed numbers with cube −1, and their 18 conjugates under permutations of coordinates.
As a consequence, a scheme is separated when the diagonal of within the scheme product of with itself is a closed immersion. Emphasizing the relative point of view, one might equivalently define a scheme to be separated if the unique morphism X → Spec ( Z ) {\displaystyle X\rightarrow {\textrm {Spec}}(\mathbb {Z} )} is separated.
The term diagonal matrix may sometimes refer to a rectangular diagonal matrix, which is an m-by-n matrix with all the entries not of the form d i,i being zero. For example: [ 1 0 0 0 4 0 0 0 − 3 0 0 0 ] or [ 1 0 0 0 0 0 4 0 0 0 0 0 − 3 0 0 ] {\displaystyle {\begin{bmatrix}1&0&0\\0&4&0\\0&0&-3\\0&0&0\\\end{bmatrix}}\quad {\text{or}}\quad ...
The image of a diagonal morphism in the category of sets, as a subset of the Cartesian product, is a relation on the domain, namely equality. For concrete categories , the diagonal morphism can be simply described by its action on elements x {\displaystyle x} of the object a {\displaystyle a} .