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Force-System Resultants and Equilibrium
Published in Richard C. Dorf, The Engineering Handbook, 2018
so that the column space is the collection of all vectors Ax, and thus the system Ax=b has a solution if and only if b is a member of the A column space. The dimension of the column space is the rank of A. The row space has the same dimension as the column space. The set of all solutions of the system Ax=0 is a subspace called the null space of A, and the dimension of this null space is the nullity of A. A fundamental result in matrix theory is the fact that, for an n×m matrix A,
Fundamentals of Systems of Differential Equations
Published in Stephen A. Wirkus, Randall J. Swift, Ryan S. Szypowski, A Course in Differential Equations with Boundary-Value Problems, 2017
Stephen A. Wirkus, Randall J. Swift, Ryan S. Szypowski
For the column space, the presence of two basic variables tells us that it will have two vectors in its basis. The pivot columns are the first and fourth and so a basis for the column space will be the first and fourth columns of the original matrix: () R(A)={[221],[210]}
Rank of a Matrix
Published in Ravi P. Agarwal, Cristina Flaut, An Introduction to Linear Algebra, 2017
Ravi P. Agarwal, Cristina Flaut
For a given matrix A ∊ Cm×n its rows (columns) generate a subspace of Cn(Cm), called the row space (column space) of A, and denoted as R(A)(C(A)) . It is clear that two row (column) equivalent matrices have the same row (column) space. The row (column) rank of the matrix A is the dimension of the row (column) space of A.
Model-matching methods and distributed control of networks consisting of a class of heterogeneous dynamic agents
Published in International Journal of Control, 2021
Eleftherios E. Vlahakis, George D. Halikias
The field of real and complex numbers is denoted by and , respectively. denotes the n-dimensional vector space over the field and denotes the set of real matrices. Let be vectors not necessarily of the same dimensions. Then, . Let , then is a diagonal matrix, being its diagonal entries. Note that if are square matrices (not necessarily of the same dimensions), is a block-diagonal matrix. We denote by the determinant of a square matrix A. The column space of a matrix is the set of all linear combinations of its columns. Let and , then denotes the column space of A and . Let . The dimension of is denoted by . The transpose of ξ is denoted by . The identity matrix of dimension is denoted by . The zero matrix is denoted by unless the dimensions are obvious in which case (part of) the subscript will be omitted. Matrix is called symmetric if . denotes the real part of . The set of complex numbers with non-positive real part is denoted by . Similarly, . denotes the Kronecker product of matrices A and B. If Ξ is symmetric, denotes the ith eigenvalue of Ξ ordered in non-decreasing order of magnitude and is the spectrum of Ξ. Matrix is called stable or Hurwitz if all its eigenvalues have negative real part, i.e., , . We will make use of the following: