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In mathematics, a block matrix or a partitioned matrix is a matrix that is interpreted as having been broken into sections called blocks or submatrices. [1] [2]Intuitively, a matrix interpreted as a block matrix can be visualized as the original matrix with a collection of horizontal and vertical lines, which break it up, or partition it, into a collection of smaller matrices.
An example of this is the function block diagram, one of five programming languages defined in part 3 of the IEC 61131 (see IEC 61131-3) standard that is highly formalized (see formal system), with strict rules for how diagrams are to be built. Directed lines are used to connect input variables to block inputs, and block outputs to output ...
This (n 1 + ⋯ + n r) × (n 1 + ⋯ + n r) square matrix, consisting of r diagonal blocks, can be compactly indicated as ,, or (,, …,,), where the i-th Jordan block is J λ i,n i. For example, the matrix = [] is a 10 × 10 Jordan matrix with a 3 × 3 block with eigenvalue 0, two 2 × 2 blocks with eigenvalue the imaginary unit i, and a 3 × ...
The Jordan block corresponding to λ is of the form λI + N, where N is a nilpotent matrix defined as N ij = δ i,j−1 (where δ is the Kronecker delta). The nilpotency of N can be exploited when calculating f(A) where f is a complex analytic function. For example, in principle the Jordan form could give a closed-form expression for the ...
Extensive use is usually made of a diagrammatic style known as the block diagram. In it the transfer function , also known as the system function or network function, is a mathematical model of the relation between the input and output based on the differential equations describing the system.
In control engineering and system identification, a state-space representation is a mathematical model of a physical system that uses state variables to track how inputs shape system behavior over time through first-order differential equations or difference equations.
In mathematics, a diagram algebra is an algebraic structure in which operations are performed using diagrams rather than traditional techniques. In particular ...
The basic idea is to try to divide a system in such a way that each block of a block diagram can be described without an "and" or "or" in the description. This exercise forces each part of the system to have a pure function. When a system is designed as pure functions, they can be reused, or replaced.
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