[JPE, May 1994 #7] Let A E R"×" be given, symmetric and positive definite. Define Ao = A, and consider the sequence of matrices defined by Ak = G,G and Ar+1 = GGR %3D where Ak GRG, is the Cholesky factorization for A. Prove that the A all have the same eigenvalues.
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please solve number 5
![5. [JPE, May 1994 #7] Let A E R"X" be given, symmetric and positive definite. Define
Ao = A, and consider the sequence of matrices defined by
Ak = GrG
and
Ak+1 = GGr
where A = GkG, is the Cholesky factorization for A. Prove that the Ak all have the
same eigenvalues.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F53e9ab80-64ca-4d50-9be9-589e5309635d%2F5b0b67c8-75c9-4cf0-a1f3-4afd46f229b8%2F4jiere7_processed.jpeg&w=3840&q=75)
Given,
It is required to prove that eigenvalues of
Now,
In the third equality we used the result that eigenvalue of are same.
We prove have the statement that are same.
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