# a) suppose that the nxn matrix A has its n eigenvalues arranged in decreasing order of absolute size, so that  >>....

a) suppose that the nxn matrix A has its n eigenvalues arranged in decreasing order of absolute size, so that  >>.... each eigenvalue has its corresponding eigenvector, x1,x2,...,xn. suppose we make some initial guess y(0) for an eigenvector. suppose, too, that y(0) can be written in terms of the actual eigenvectors in the form y(0)=alpha1.x1 +alpha2.x2 +...+alpha(n).x(n), where alpha1, alpha2, alpha(n) are constants. by considering the "power method" type iteration y(k+1)=Ay(k) argue that (see attached image)

b) from an nxn matrix A, with eigenvalues and eigenvectors in part a, show that inverse matrix A^-1 has the same eigenvectors as A, but that its eigenvalues are 1/lamda1, 1/lamda2,...,1/lamda(n). what will the power method do, if applied to this inverse matrix A^-1

c)the nxn matrix B is defined to be similar to nxn matrix A if B=P^(-1)AP. here P is a non-singular nxn matrix. show that B has the same eigenvalues as A.

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4. (a) Suppose that the (nxn) matrix A has its n decreasing order of absolute size, so that |2,| > |2,| >... >|2|. Each eigenvalue has its corresponding eigenvector, x, , x2 , ..., x, Suppose we make some initial guess y0) for an eigenvector. Suppose, too, that y(0) can be written in terms of the actual eigenvectors in the form eigenvalues arranged in у 3 а, х, +а,х, +...+a,х, where a a2, ... , a, are constants. By considering the "power method" type iteration yA+1) = Ay(*), argue that (k) уt-) (k+1) (k+1) and as yy) (k) (k) (k+1) [10 points] (b) For an (nxn) matrix A, with eigenvalues and eigenvectors as in part(a), show that the inverse matrix A1 has the same eigenvectors as A , but that 1 its eigenvalues are What will the power method do, if applied to this inverse matrix A1 ? [5 points] (c) The (nxn) matrix B is defined to be similar to B PAP. Here, P is a non-singular (nxn) matrix. Show that B has the same eigenvalues as A (nxn) matrix A if [5 points

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