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The matrix A is said to represent the linear map f, and A is called the transformation matrix of f. For example, the 2×2 matrix = [] can be viewed as the transform of the unit square into a parallelogram with vertices at (0, 0), (a, b), (a + c, b + d), and (c, d).
The definition of matrix multiplication is that if C = AB for an n × m matrix A and an m × p matrix B, then C is an n × p matrix with entries = =. From this, a simple algorithm can be constructed which loops over the indices i from 1 through n and j from 1 through p, computing the above using a nested loop:
As a direct consequence of simultaneous triangulizability, the eigenvalues of two commuting complex matrices A, B with their algebraic multiplicities (the multisets of roots of their characteristic polynomials) can be matched up as in such a way that the multiset of eigenvalues of any polynomial (,) in the two matrices is the multiset of the ...
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 first and the number of columns of the second matrix. The product of matrices A and B is denoted as AB. [1]
where (,) denotes the set of 2 by 2 complex matrices, is an injective real linear map (by considering diffeomorphic to and (,) diffeomorphic to ). Hence, the restriction of φ to the 3-sphere (since modulus is 1), denoted S 3 , is an embedding of the 3-sphere onto a compact submanifold of M ( 2 , C ) {\displaystyle \operatorname {M ...
In linear algebra, two n-by-n matrices A and B are called similar if there exists an invertible n-by-n matrix P such that =. Similar matrices represent the same linear map under two (possibly) different bases, with P being the change-of-basis matrix. [1] [2]
This reduces the number of matrix additions and subtractions from 18 to 15. The number of matrix multiplications is still 7, and the asymptotic complexity is the same. [6] The algorithm was further optimised in 2017, [7] reducing the number of matrix additions per step to 12 while maintaining the number of matrix multiplications, and again in ...
Specht's theorem: Two matrices A and B are unitarily equivalent if and only if tr W(A, A*) = tr W(B, B*) for all words W. [4] The theorem gives an infinite number of trace identities, but it can be reduced to a finite subset. Let n denote the size of the matrices A and B. For the case n = 2, the following three conditions are sufficient: [5]