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Similarity is an equivalence relation on the space of square matrices. Because matrices are similar if and only if they represent the same linear operator with respect to (possibly) different bases, similar matrices share all properties of their shared underlying operator: Rank; Characteristic polynomial, and attributes that can be derived from it:
In linear algebra, two rectangular m-by-n matrices A and B are called equivalent if = for some invertible n-by-n matrix P and some invertible m-by-m matrix Q.Equivalent matrices represent the same linear transformation V → W under two different choices of a pair of bases of V and W, with P and Q being the change of basis matrices in V and W respectively.
If a 2 x 2 real matrix has zero trace, its square is a diagonal matrix. The trace of a 2 × 2 complex matrix is used to classify Möbius transformations. First, the matrix is normalized to make its determinant equal to one. Then, if the square of the trace is 4, the corresponding transformation is parabolic.
In mathematics, every analytic function can be used for defining a matrix function that maps square matrices with complex entries to square matrices of the same size.. This is used for defining the exponential of a matrix, which is involved in the closed-form solution of systems of linear differential equations.
The spectral theorem for normal matrices is a special case of the more general Schur decomposition which holds for all square matrices. Let A be a square matrix. Then by Schur decomposition it is unitary similar to an upper-triangular matrix, say, B. If A is normal, so is B.
Given an arbitrary square matrix, the elementary divisors used in the construction of the Jordan normal form do not exist over F[X], so the invariant factors f i as given above must be used instead. The last of these factors f k is then the minimal polynomial, which all the invariant factors therefore divide, and the product of the invariant ...
A square matrix derived by applying an elementary row operation to the identity matrix. Equivalent matrix: A matrix that can be derived from another matrix through a sequence of elementary row or column operations. Frobenius matrix: A square matrix in the form of an identity matrix but with arbitrary entries in one column below the main diagonal.
In general, a square complex matrix A is similar to a block diagonal matrix = [] where each block J i is a square matrix of the form = []. So there exists an invertible matrix P such that P −1 AP = J is such that the only non-zero entries of J are on the diagonal and the superdiagonal.