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An orthogonal matrix Q is necessarily invertible (with inverse Q−1 = QT ), unitary ( Q−1 = Q∗ ), where Q∗ is the Hermitian adjoint ( conjugate transpose) of Q, and therefore normal ( Q∗Q = QQ∗) over the real numbers. The determinant of any orthogonal matrix is either +1 or −1. As a linear transformation, an orthogonal matrix ...
In mathematics, a matrix ( pl.: matrices) is a rectangular array or table of numbers, symbols, or expressions, arranged in rows and columns, which is used to represent a mathematical object or property of such an object. For example, is a matrix with two rows and three columns. This is often referred to as a "two by three matrix", a " matrix ...
In mathematics, particularly in linear algebra, matrix multiplication is a binary operation that produces a matrix from two matrices. For matrix multiplication, the number of columns in the first matrix must be equal to the number of rows in the second matrix. The resulting matrix, known as the matrix product, has the number of rows of the ...
In mathematics, the determinant is a scalar value that is a certain function of the entries of a square matrix. The determinant of a matrix A is commonly denoted det (A), det A, or |A|. Its value characterizes some properties of the matrix and the linear map represented, on a given basis, by the matrix. In particular, the determinant is nonzero ...
Invertible matrix. In linear algebra, an n -by- n square matrix A is called invertible (also nonsingular, nondegenerate or rarely regular) if there exists an n -by- n square matrix B such that. where In denotes the n -by- n identity matrix and the multiplication used is ordinary matrix multiplication. [1]
The matrix may be regarded as a diagonal matrix that has been re-expressed in coordinates of the (eigenvectors) basis . Put differently, applying M {\displaystyle M} to some vector z , giving M z , is the same as changing the basis to the eigenvector coordinate system using P −1 , giving P −1 z , applying the stretching transformation D to ...
If A is an m × n matrix and A T is its transpose, then the result of matrix multiplication with these two matrices gives two square matrices: A A T is m × m and A T A is n × n. Furthermore, these products are symmetric matrices. Indeed, the matrix product A A T has entries that are the inner product of a row of A with a column of A T.
The exponential of X, denoted by eX or exp (X), is the n×n matrix given by the power series. where is defined to be the identity matrix with the same dimensions as . [1] The series always converges, so the exponential of X is well-defined. Equivalently, where I is the n×n identity matrix. When X is an n×n diagonal matrix then exp (X) will be ...