Using Householder’s transformation, reduce the following matrices in tridiagonal form and find all the eigen values and eigen vectors. Write the strum sequence.

     \(\left[ \begin{array}{cc} 1&\sqrt{2}&\sqrt{2} &2\\ \sqrt{2}&-\sqrt{2}&-1&\sqrt{2}\\\sqrt{2} &-1&\sqrt{2}&\sqrt{2}\\ 2&\sqrt{2}&\sqrt{2}&-3 \end{array} \right]\)

 

If the orthogonal transform is of the form \(p=1-2WW^T\) where \(W\) is a column vector such that \(W^TW=1\). Prove that \(P\) is symmetric and orthogonal.

Write the strum sequence and find the eigen values of following matrix:

    \(\left[ \begin{array}{cc} 1&-1&0&0&0&0 \\ 1&1&-1&0&0&0 \\ 0&1&1&-1&0&0 \\ 0&0&1&1&-1&0 \\0&0&0&1&1&-1 \\ 0&0&0&0&1&1\end{array} \right]\)

Use LR and QR both the methods to find the eigen values of the following matrix:

    \(A =\left[ \begin{array}{cc} 3&2 \\ 1&4 \end{array} \right]\)

Let \(R\) be a ring in which \(a^2=a, \forall a \in R\) (such a ring is called a Boolean ring). Show that \(2a=0, \forall a\in R\) , and \(R\) is commutative. Further show that the only Boolean ring that is an integral domain is \(\mathbb{Z_2}\).

Let \(R\) be a ring and \(a,b \in R\)such that \(ab=ba\) . Prove that for any positive integer  \((a+b)^n =a^n +\left( \begin{array}{c} n \\ 1 \end{array} \right)a^{n-1}b+ \cdots +\left( \begin{array}{c} n \\ n-1 \end{array} \right) ab^{n-1}+b^n\) .

Let \(d\) be any integer. Prove that \(\mathbb{Z}[\sqrt{d}] =\{a+b\sqrt{d} \; |a.b \in \mathbb{Z} \}\) is an integral domain and \(\mathbb{Z}[\sqrt{d}] =\{a+b\sqrt{d} \; |a.b \in \mathbb{Z} \}\) 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 \(1^\prime\) but \(R\) has either no identity or the identity of \(R\) is different from \(1^\prime\) , then show that \(1^\prime\) 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 \(R >0\)and infinite if char \(R=0\) .

If \(a\) is a nilpotent element in a ring \(R\) with identity 1, then show that  \(1-a\) is a unit in \(R\). Further, show that if \(R\) is commutative, then \(1-ab\) is a unit for all  \(b \in R\) .