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The exponential of a matrix A is defined by =!. Given a matrix B, another matrix A is said to be a matrix logarithm of B if e A = B.. Because the exponential function is not bijective for complex numbers (e.g. = =), numbers can have multiple complex logarithms, and as a consequence of this, some matrices may have more than one logarithm, as explained below.
The following tables list the computational complexity of various algorithms for common mathematical operations. Here, complexity refers to the time complexity of performing computations on a multitape Turing machine. [1] See big O notation for an explanation of the notation used. Note: Due to the variety of multiplication algorithms, below ...
If this is the case, then the matrix B is uniquely determined by A, and is called the (multiplicative) inverse of A, denoted by A −1. Matrix inversion is the process of finding the matrix which when multiplied by the original matrix gives the identity matrix. [2] Over a field, a square matrix that is not invertible is called singular or ...
Matrix (mathematics) An m × n matrix: the m rows are horizontal and the n columns are vertical. Each element of a matrix is often denoted by a variable with two subscripts. For example, a2,1 represents the element at the second row and first column of the matrix. In mathematics, a matrix (pl.: matrices) is a rectangular array or table of ...
Vectorization (mathematics) In mathematics, especially in linear algebra and matrix theory, the vectorization of a matrix is a linear transformation which converts the matrix into a vector. Specifically, the vectorization of a m × n matrix A, denoted vec (A), is the mn × 1 column vector obtained by stacking the columns of the matrix A on top ...
Trace (linear algebra) In linear algebra, the trace of a square matrix A, denoted tr (A), [1] is defined to be the sum of elements on the main diagonal (from the upper left to the lower right) of A. The trace is only defined for a square matrix (n × n). In mathematical physics texts, if tr (A) = 0 then the matrix is said to be traceless.
Minor (linear algebra) In linear algebra, a minor of a matrix A is the determinant of some smaller square matrix, cut down from A by removing one or more of its rows and columns. Minors obtained by removing just one row and one column from square matrices (first minors) are required for calculating matrix cofactors, which in turn are useful for ...
In vector calculus, the Jacobian matrix (/ dʒəˈkoʊbiən /, [1][2][3] / dʒɪ -, jɪ -/) of a vector-valued function of several variables is the matrix of all its first-order partial derivatives. When this matrix is square, that is, when the function takes the same number of variables as input as the number of vector components of its output ...
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