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eigen - Coggle Diagram
eigen
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A is invertible, eigenvalues is non zero
A is singular, A has zero eigenvalues
λ = 0, eigenvectors are null space vectors of A
λ != 0, eigenvectors are null space vectors of |A - λ |
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Sum of Eigenvalues
Product of Eigenvalues
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|A| = λ1λ2...λN
If A is singular, then at least one λ is 0
λs are diagonal entries of A's triangular form
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all diagonal matrices have N independent eigenvectors
triangular matrices may or may not have N independent eigenvectors
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Given N independent eigenvector x1,x2,...xn
Any vector v = c1X1 + c2X2 + ... + cnXn
A^100 v => Λ^100 S [c1,c2,..cn]
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If A = A^t
A = LU = U^tU => L = U^t
det(L) = 1
det(A) = det(U^t)det(U) = det(U)
det(A) = dd1dd2...ddn (diagonal elems of U)
det|A| = det|D| doesn't mean their eigenvalues are same
A is singular, then A has a eigenvalue 0
tr(A) = λ1 + λ2 + ... + λn
det(A) =