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Lemma 1. ′ =, where ′ is the differential of . This equation means that the differential of , evaluated at the identity matrix, is equal to the trace.The differential ′ is a linear operator that maps an n × n matrix to a real number.
The all-ones matrix arises in the mathematical field of combinatorics, particularly involving the application of algebraic methods to graph theory.For example, if A is the adjacency matrix of an n-vertex undirected graph G, and J is the all-ones matrix of the same dimension, then G is a regular graph if and only if AJ = JA. [7]
The th column of an identity matrix is the unit vector, a vector whose th entry is 1 and 0 elsewhere. The determinant of the identity matrix is 1, and its trace is . The identity matrix is the only idempotent matrix with non-zero determinant. That is, it is the only matrix such that:
For example, if A is a 3-by-0 matrix and B is a 0-by-3 matrix, then AB is the 3-by-3 zero matrix corresponding to the null map from a 3-dimensional space V to itself, while BA is a 0-by-0 matrix. There is no common notation for empty matrices, but most computer algebra systems allow creating and computing with them.
The solution is the product . [3] This intuitively makes sense because an orthogonal matrix would have the decomposition where is the identity matrix, so that if = then the product = amounts to replacing the singular values with ones.
The sum of the first three terms of this equation, namely + + = (/ /) (), is the quadratic form associated with the equation, and the matrix = (/ /) is called the matrix of the quadratic form. The trace and determinant of A 33 {\displaystyle A_{33}} are both invariant with respect to rotation of axes and translation of the plane (movement of ...
In the mathematical field of algebraic graph theory, the degree matrix of an undirected graph is a diagonal matrix which contains information about the degree of each vertex—that is, the number of edges attached to each vertex. [1]
The characteristic equation, also known as the determinantal equation, [1] [2] [3] is the equation obtained by equating the characteristic polynomial to zero. In spectral graph theory , the characteristic polynomial of a graph is the characteristic polynomial of its adjacency matrix .