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In this case, the most important component shape to consider is the rectangle. [1] Rectangular partitions have many applications. In VLSI design, it is necessary to decompose masks into the simpler shapes available in lithographic pattern generators, and similar mask decomposition problems also arise in DNA microarray design.
A ‘quasimatrix’ is, like a matrix, a rectangular scheme whose elements are indexed, but one discrete index is replaced by a continuous index. Likewise, a ‘cmatrix’, is continuous in both indices. As an example of a cmatrix, one can think of the kernel of an integral operator.
Specifically, the singular value decomposition of an complex matrix is a factorization of the form =, where is an complex unitary matrix, is an rectangular diagonal matrix with non-negative real numbers on the diagonal, is an complex unitary matrix, and is the conjugate transpose of . Such decomposition ...
The simplest rectilinear polygon is an axis-aligned rectangle - a rectangle with 2 sides parallel to the x axis and 2 sides parallel to the y axis. See also: Minimum bounding rectangle . A golygon is a rectilinear polygon whose side lengths in sequence are consecutive integers.
More generally, we can factor a complex m×n matrix A, with m ≥ n, as the product of an m×m unitary matrix Q and an m×n upper triangular matrix R.As the bottom (m−n) rows of an m×n upper triangular matrix consist entirely of zeroes, it is often useful to partition R, or both R and Q:
If P x is a rectangle of sides a · x and a · (1/x) and Q is a square of side length a, then P x and Q are equidecomposable for every x > 0. An upper bound for σ( P x , Q ) is given by [ 3 ] σ ( P x , Q ) ≤ 2 + ⌈ x 2 − 1 ⌉ , for x ≥ 1. {\displaystyle \sigma (P_{x},Q)\leq 2+\left\lceil {\sqrt {x^{2}-1}}\right\rceil ,\quad {\text{for ...
Let A be a square n × n matrix with n linearly independent eigenvectors q i (where i = 1, ..., n).Then A can be factored as = where Q is the square n × n matrix whose i th column is the eigenvector q i of A, and Λ is the diagonal matrix whose diagonal elements are the corresponding eigenvalues, Λ ii = λ i.
The subdivided regions may be square or rectangular, or may have arbitrary shapes. This data structure was named a quadtree by Raphael Finkel and J.L. Bentley in 1974. [1] A similar partitioning is also known as a Q-tree. All forms of quadtrees share some common features: They decompose space into adaptable cells.