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In quantum mechanics, a density matrix (or density operator) is a matrix that describes an ensemble [1] of physical systems as quantum states (even if the ensemble contains only one system). It allows for the calculation of the probabilities of the outcomes of any measurements performed upon the systems of the ensemble using the Born rule .
Suppose a zero-sum game has a payoff matrix M where element M i,j is the payoff obtained when the minimizing player chooses pure strategy i and the maximizing player chooses pure strategy j (i.e. the player trying to minimize the payoff chooses the row and the player trying to maximize the payoff chooses the column).
A matrix, has its column space depicted as the green line. The projection of some vector onto the column space of is the vector . From the figure, it is clear that the closest point from the vector onto the column space of , is , and is one where we can draw a line orthogonal to the column space of .
In other words, the matrix of the combined transformation A followed by B is simply the product of the individual matrices. When A is an invertible matrix there is a matrix A −1 that represents a transformation that "undoes" A since its composition with A is the identity matrix. In some practical applications, inversion can be computed using ...
and it allows to polynomial-time reduce the computation of the permanent of an n×n-matrix with a subset of k or k − 1 rows expressible as linear combinations of another (disjoint) subset of k rows to the computation of the permanent of an (n − k)×(n − k)- or (n − k + 1)×(n − k + 1)-matrix correspondingly, hence having introduced a ...
To this end, we will construct a 20x20 matrix where the (,) th entry is equal to the probability of the th amino acid being transformed into the th amino acid in a certain amount of evolutionary time. There are many different ways to construct such a matrix, called a substitution matrix. Here are the most commonly used ones:
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Lloyd Shapley in 2012. The Shapley value is a solution concept in cooperative game theory.It was named in honor of Lloyd Shapley, who introduced it in 1951 and won the Nobel Memorial Prize in Economic Sciences for it in 2012.