MTH603 Quiz 1 Solution and Discussion
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while using jacobi method fot the matrix A=[1 1/4 1/4 1/4 1/3 1/2 1/3 1/2 1/5]
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Let A be an n × n matrix. Then λ = 0 is an eigenvalue of A if and only if there exists a non-zero vector v ∈ Rn such that Av = λv = 0. In other words, 0 is an eigenvalue of A if and only if the vector equation Ax = 0 has a non-zero solution x ∈ Rn.
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an eigenvector v is said to be normalized if the coordinate of largest magnitude is equal to aero?
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Two similar matrices have the same eigenvalues, even though they will usually have different eigenvectors. Said more precisely, if B = Ai’AJ. I and x is an eigenvector of A, then M’x is an eigenvector of B = M’AM. So, A1’x is an eigenvector for B, with eigenvalue ).
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b) The eigenvalues of a real symmetric matrix need not be positive. They can be positive, negative, or even zero, depending on the elements and the specific structure of the matrix.
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Jacobian Method in Matrix Form
Let the n system of linear equations be Ax = b. Let us decompose matrix A into a diagonal component D and remainder R such that A = D + R. Iteratively the solution will be obtained using the below equation. -
for a function; y=f(x), if y0, y1 are 2,3
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how many eigenvalues will exit corresponding to the function exp(ax)