All Questions page - 70

Prove that \(S_n\) is generated by the set of transpositions \(\{(1 \; 2),(2 \;3), \cdots ,(n-1 \; n ) \}\) .

Using Jacobi method and cyclic Jacobi method, find all the eigen values and corresponding eigen vectors of the following matrices:

   (a) \(\left[\begin{array}{cols} 2&1.414&4 \\1.414&6&1.414 \\ 1&1.414 &2 \end{array}\right] \)

Using Jacobi method and cyclic Jacobi method, find all the eigen values and corresponding eigen vectors of the following matrices:

   (a)  \(\left[ \begin{array}{cols} 2&3&1\\3&2&2\\ 1&2&1 \end{array}\right] \)

 In case of, (a) iterate till off diagonal elements in magnitude are less than 0.0005.

What are the advantages of Given’s method over the Jacobi’s method?

Transform the following matrices to tridiagonal system by Given’s method:

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

Transform the following matrices to tridiagonal system by Given’s method:

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

Transform the following matrices to tridiagonal system by Given’s method and find the smallest eigen value correct to 2d places using the Newton-Raphson method. Also find the eigen vector corresponding to that eigen value.

    \(\left[\begin{array}{cols} 1&2&2\\2&1&2\\2&2&1 \end{array}\right] \)

Transform the following matrices to tridiagonal system by Given’s method and find the smallest eigen value correct to 2d places using the Newton-Raphson method. Also find the eigen vector corresponding to that eigen value.

     \(\left[\begin{array}{cols} 2&1.414&4 \\ 1.414&6&1.414 \\ 1&1.414&2 \end{array}\right] \)

How many rotations are required to reduce  above matrices to tridiagonal form.

  \(\left[\begin{array}{cols} 1&2&2 \\2&1&2 \\2&2&1 \end{array}\right] \)

How many rotations are required to reduce each of above matrices to tridiagonal form.

     \(\left[ \begin{array}{cc} 2& 1.414&4 \\ 1.414 &6&1.414 \\ 1&1.414&2 \end{array} \right]\)

How the Householder method is better than Given’s method?

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

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

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 \(\mathbb{Z_2}\).