Browse & Filter

Show that the eigenvectors \(x_1, x_2, \ldots, x_n\) of an operator \(A\) corresponding to distinct eigenvalues \(\lambda_1, \lambda_2, \ldots\lambda_n \) are linearly independent.

In \(M^3(C)\) consisting of all \(3\times3\) complex matrices \begin{equation}   A=  \begin{pmatrix}         A_{11} & A_{12} & A_{13} \\         A_{21} & A_{22} & A_{23} \\           A_{31} & A_{32} & A_{33}   \end{pmatrix} \end{equation} which of the following are linear functionals?

Find the matrix representation of the  operators \(A\) on the vector space of all  \(2\times 2\) real matrices, where
\[ A \begin{pmatrix} a & b \\c & d\end{pmatrix} =
    \begin{pmatrix}1 & 1\\1& 1 \end{pmatrix}
    \begin{pmatrix} a & b \\c & d\end{pmatrix},\]
in the basis
 \begin{equation*}
  e_1 = \begin{pmatrix}
          1 & 0\\0& 0&0
        \end{pmatrix};\quad
  e_2 = \begin{pmatrix}
         0 & 1 \\ 0 & 0
        \end{pmatrix};\quad
  e_3 = \begin{pmatrix}
         0 & 0 \\ 1 & 0
         \end{pmatrix};\quad
  e_4 = \begin{pmatrix}
          0 & 0 \\
          0 & 1
        \end{pmatrix}.
\end{equation*}

1. Vector Spaces with Inner ProductInner product; Orthogonality and Gram Schmidt Procedure; Cauchy Schwarz and Triangle Inequalities.

2. Linear Operators in Inner Product SpacesLinear operators; Adjoint of a linear operator; Hermitian operator; Unitary operators.

3. Complete Orthonormal SetsComplete orthonormal sets; Setting up a representation in o.n basis; Dirac Bra and Ket Notation.

Vector Spaces with Inner Product

1. Inner product; Orthogonality and Gram Schmidt Procedure; Cauchy Schwarz and Triangle Inequalities.

2. Linear Operators in Inner Product SpacesLinear operators; Adjoint of a linear operator; Hermitian operator; Unitary operators.

3. Complete Orthonormal SetsComplete orthonormal sets; Setting up a representation in o.n basis; Dirac Bra and Ket Notation.

Vectors are needed for almost all areas of Physics. This is a quick reminder of vector  analysis for those  who have had one exposure to the area.

Find the matrix representation of the  operators \(A\) on the space of all polynomials\(\{p(t)\}\), where
\[ A p(t) = p(t+1)\]
in the bases
(i) \(1,t,t^2 \)\\
(ii)\( 1, t+t^2 , t-t^2 \)
How are the two representations related?

Find the Jordan canonical form of the following matrices.

1. \(\begin{pmatrix} 0 & 3 \\ 2 & -1\end{pmatrix}\)
2. \( \begin{pmatrix} 3 &2 &4 \\ 2 & 0 & 2 \\ 4& 2 & 3 \end{pmatrix}\)
3. \(\begin{pmatrix}0 & 1 & 0 \\ 0& 0 & 1\\ -1 & 1 &1 \end{pmatrix}\)

Prove that the following inequalities hold in every inner product space

1. \( \Big| x-y \Big| \le \Vert x\Vert + \Vert y\Vert \);
2. \( \Big|x-y\Big| \ge \Vert x\Vert - \Vert y\Vert \).
3. \( \Big| \sum_{i=1}^n x_i \Big| \le \sum_{i=1}^n \Vert x_i\Vert \);