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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:
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 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.
Square D is an American manufacturer of electrical equipment headquartered in Andover, Massachusetts. Square D is a flagship brand of Schneider Electric , which acquired the company in 1991. The company was listed on the New York Stock Exchange for 55 years prior to its acquisition without reporting financial loss in any calendar quarter ...
In control system theory, and various branches of engineering, a transfer function matrix, or just transfer matrix is a generalisation of the transfer functions of single-input single-output (SISO) systems to multiple-input and multiple-output (MIMO) systems. [1] The matrix relates the outputs of the system to its inputs.
If instead A is a complex square matrix, then there is a decomposition A = QR where Q is a unitary matrix (so the conjugate transpose † =). If A has n linearly independent columns, then the first n columns of Q form an orthonormal basis for the column space of A .
For matrix-matrix exponentials, there is a distinction between the left exponential Y X and the right exponential X Y, because the multiplication operator for matrix-to-matrix is not commutative. Moreover, If X is normal and non-singular, then X Y and Y X have the same set of eigenvalues. If X is normal and non-singular, Y is normal, and XY ...
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 ...