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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: Determinant; Trace; Eigenvalues, and their algebraic multiplicities
It is represented on "old" bases of V and W by a m×n matrix M. A change of bases is defined by an m×m change-of-basis matrix P for V, and an n×n change-of-basis matrix Q for W. On the "new" bases, the matrix of T is . This is a straightforward consequence of the change-of-basis formula.
In Einstein notation (implicit summation over repeated index), contravariant components are denoted with upper indices as in = A covector or cotangent vector has components that co-vary with a change of basis in the corresponding (initial) vector space. That is, the components must be transformed by the same matrix as the change of basis matrix ...
Multi-index notation is a mathematical notation that simplifies formulas used in multivariable calculus, partial differential equations and the theory of ...
This is known as base conversion. The logarithmic change-of-base formula, one of the logarithmic identities used frequently in algebra and calculus. The method for changing between polynomial and normal bases, and similar transformations, for purposes of coding theory and cryptography. Construction of the fiber product of schemes, in algebraic ...
It is common convention to use greek indices when writing expressions involving tensors in Minkowski space, while Latin indices are reserved for Euclidean space. Well-formulated expressions are constrained by the rules of Einstein summation: any index may appear at most twice and furthermore a raised index must contract with a lowered index ...
Graphs of y = b x for various bases b: base 10, base e, base 2, base 1 / 2 . Each curve passes through the point (0, 1) because any nonzero number raised to the power of 0 is 1. At x = 1, the value of y equals the base because any number raised to the power of 1 is the number itself.
Ordering of indices is significant, even when of differing variance. However, when it is understood that no indices will be raised or lowered while retaining the base symbol, covariant indices are sometimes placed below contravariant indices for notational convenience (e.g. with the generalized Kronecker delta).