Matrix nonsingular

Nonsingular matrix A matrix is nonsingular if its determinant is not zero. 

Before this is done, however, a certain paradox needs to be discussed briefly. Given a matrix A, and a nonsingular matrix V, it is known that A, and V XA V, have the same characteristic polynomial, and the two matrices are said to be similar. Among all matrices similar to a given matrix A, there are matrices of the form... 

Nonlinear systems, 78 analytical methods, 349 Nonlinearities, nonanalytic, 383,389 Nonsingular matrix, 57 Nonunitary groups, 725 as co-representations, 731 representation theory, 728 structure of, 727 Nonunitary point groups, 737 No-particle state. 540,708 expectation value of current operator, 587 out, 586... 

We first do a demonstration of similarity transform. For a nonsingular matrix A with distinct eigenvalues, we can find a nonsingular (modal) matrix P such that the matrix A can be transformed into a diagonal made up of its eigenvalues. This is one useful technique in decoupling a set of differential equations. 

Note that the denominator of (A. 17), the determinant of A = A, is a scalar. If A = 0, the inverse does not exist. A square matrix with determinant equal to zero is called a singular matrix. Conversely, for a nonsingular matrix A, det A 0. 

If G is an (m x g) matrix of rank k and Ug denotes the column-echelon form of G, then a nonsingular matrix Eq exists such that... 

A nonsingular matrix can be expressed as a product of elementary matrices... 

Also, (2) follows immediately since the column-echelon form of a nonsingular matrix is the unit matrix. ... 

Proof. Inserting the expression for - given in Lemma 3 into the left-hand side of (34) and multiplying the equation thus obtained by the inverse of the nonsingular matrix H we arrive at the system of matrix equations, whose left-hand sides are the linear combinations of the linearly independent matrices E, Syield system of Eqs. (39). The assertion is proved. 

Theorem 4.1. LetA(t) be periodic of period T Then if 4>(f) is a fundamental matrix, so is (0 = t + T). Corresponding to any fundamental matrix (f) there exists a periodic nonsingular matrix P(t) of period T and a constant matrix B such that... 

Equation (3.3) can be simplified further by the change of variables q = T p, where T is the transpose of a nonsingular matrix T to be determined shortly. Introducing this change in (3.3) results in... 

The Gaussian algorithm described in Section A.4 transforms the matrix A into an upper triangular matrix U by operations equivalent to premultiplication of A by a nonsingular matrix. Denoting the latter matrix by one obtains the representation... 

Here Vn is a nonsingular matrix whose last column is special, with elements... 

Thus square matrices can always be multiplied in any order. Also, we have lA = A for any matrix A, which also implies that an identity matrix raised to any exponent also gives I. A nonsingular matrix has an inverse matrix, denoted by A , with the property A A = AA = I. 

It is important to discuss the concept of uniqueness at this point. The principal components are unique but are not unique in providing a basis for the plane of closest fit. This plane can also be described using another basis, e.g., P can be rotated by Z (where Z is an R x R nonsingular matrix). Then upon using (7J) 1 to counterrotate T, the solution TP does not change TP = T(Z ) z P = T(PZ) = TP, where P is the new basis and T are the scores with respect to the new basis. This property is known as rotational freedom [Harman 1967], Summarizing, the plane found is unique, but not its basis vectors.1... 

Because A and A span the same subspace (i.e. the ranges of Xi and X2), there exists a nonsingular matrix S (R x R) such that... 

If B is a nonsingular matrix of order n with n distinct eigenvalues, then... 

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