Using Householder’s transformation, reduce the following matrices in tridiagonal form and find all the eigen values and eigen vectors. Write the strum sequence.
[122212221]
Adv.
[1√2√22√2−√2−1√2√2−1√2√22√2√2−3]
If the orthogonal transform is of the form p=1−2WWT where W is a column vector such that WTW=1. Prove that P is symmetric and orthogonal.
Write the strum sequence and find the eigen values of following matrix:
[1−1000011−1000011−1000011−1000011−1000011]
Use LR and QR both the methods to find the eigen values of the following matrix:
A=[3214]
Let R be a ring in which a2=a,∀a∈R (such a ring is called a Boolean ring). Show that 2a=0,∀a∈R , and R is commutative. Further show that the only Boolean ring that is an integral domain is Z2.
Let R be a ring and a,b∈Rsuch that ab=ba . Prove that for any positive integer (a+b)n=an+(n1)an−1b+⋯+(nn−1)abn−1+bn .
Let d be any integer. Prove that Z[√d]={a+b√d|a.b∈Z} is an integral domain and Z[√d]={a+b√d|a.b∈Z} is a field.
Show that any ring with 6 elements is commutative.
If a non-zero subring S of a ring R has an identity 1′ but R has either no identity or the identity of R is different from 1′ , then show that 1′ is a zero-divisor in R .
Prove that all the non-zero elements in an integral domain have same additive order, which is the characteristic of R if char R>0and infinite if char R=0 .
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