Normal modes for the general equation Mx = −Kx
... Some extra notes on orthogonality: One way of thinking about the new orthogonality rules (19,20) is this: we are using the eigenvectors as our basis vectors, but they are not orthogonal; when our basis vectors are not orthogonal, we need to distinguish between the original vector space, in which the ...
... Some extra notes on orthogonality: One way of thinking about the new orthogonality rules (19,20) is this: we are using the eigenvectors as our basis vectors, but they are not orthogonal; when our basis vectors are not orthogonal, we need to distinguish between the original vector space, in which the ...
A proof of the Jordan normal form theorem
... Av = µv for some 0 6= v ∈ Ker(A − λ Id)k , then (A − λ Id)v = (µ − λ)v, and 0 = (A − λ Id)k v = (µ − λ)k v, so µ = λ), and the dimension of Im(A − λ Id)k is less than dim V (if λ is an eigenvalue of A), so we can apply the induction hypothesis for A acting on the vector space V 0 = Im(A − λ Id)k . T ...
... Av = µv for some 0 6= v ∈ Ker(A − λ Id)k , then (A − λ Id)v = (µ − λ)v, and 0 = (A − λ Id)k v = (µ − λ)k v, so µ = λ), and the dimension of Im(A − λ Id)k is less than dim V (if λ is an eigenvalue of A), so we can apply the induction hypothesis for A acting on the vector space V 0 = Im(A − λ Id)k . T ...
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... defined: addition, and multiplication by scalars. If the following axioms are satisfied by all objects u, v, w in V and all scalars k and m, then we call V a vector space and we call the objects in V vectors 1. If u and v are objects in V, then u + v is in V. ...
... defined: addition, and multiplication by scalars. If the following axioms are satisfied by all objects u, v, w in V and all scalars k and m, then we call V a vector space and we call the objects in V vectors 1. If u and v are objects in V, then u + v is in V. ...
Common Patterns and Pitfalls for Implementing Algorithms in Spark
... Or simply use the Spark API doubleRDD.variance() ...
... Or simply use the Spark API doubleRDD.variance() ...