Addressing #16

src/6_eigenvectors_QM.md

@@ -47,7 +47,7 @@ of vector space). We can express the previous equation in terms of its component
 assuming as usual some specific choice of basis, by using
 the rules of matrix multiplication:
 
-!!! tip "Eigenvalue equation: Eigenvalue and Eigenvector"
+!!! info "Eigenvalue equation: Eigenvalue and Eigenvector"
     $$
     \sum_{j=1}^n A_{ij} v_j = \lambda v_i \, .
     $$
@@ -57,7 +57,7 @@ the rules of matrix multiplication:
 !!! warning "Number of solutions"
     In general, there will be multiple solutions to the eigenvalue equation $A \vec{v} =\lambda \vec{v}$, each one characterised by an specific eigenvalue and eigenvectors. Note that in some cases one has  *degenerate solutions*, whereby a given matrix has two or more eigenvectors that are equal.
 
-!!! info "Characteristic equation:"
+!!! tip "Characteristic equation:"
     In order to determine the eigenvalues of the matrix $A$, we need to evaluate the solutions of the so-called *characteristic equation*
     of the matrix $A$, defined as
     $$
@@ -225,43 +225,35 @@ The set of all the eigenvalues of an operator is called *eigenvalue spectrum* of
 
 ##Problems
 
-**1)** *Eigenvalues and Eigenvectors* 
+1. *Eigenvalues and Eigenvectors* 
 
-Find the characteristic polynomial and eigenvalues for each of the following matrices,
+    Find the characteristic polynomial and eigenvalues for each of the following matrices,
+    $$A=\begin{pmatrix} 5&3\\2&10 \end{pmatrix}\,  \quad
+    B=\begin{pmatrix} 7i&-1\\2&6i \end{pmatrix} \, \quad C=\begin{pmatrix} 2&0&-1\\0&3&1\\1&0&4 \end{pmatrix}$$
 
-$$A=\begin{pmatrix} 5&3\\2&10 \end{pmatrix}\,  \quad
-B=\begin{pmatrix} 7i&-1\\2&6i \end{pmatrix} \, \quad C=\begin{pmatrix} 2&0&-1\\0&3&1\\1&0&4 \end{pmatrix}$$
+2. The Hamiltonian for a two-state system is given by 
+ 
+    $$H=\begin{pmatrix} \omega_1&\omega_2\\  \omega_2&\omega_1\end{pmatrix}$$
+    A basis for this system is 
+    $$|{0}\rangle=\begin{pmatrix}1\\0  \end{pmatrix}\, ,\quad|{1}\rangle=\begin{pmatrix}0\\1  \end{pmatrix}$$
+    Find the eigenvalues and eigenvectors of the Hamiltonian $H$, and express the eigenvectors in terms of $\{|0 \rangle,|1\rangle \}$
 
-**2)** The Hamiltonian for a two-state system is given by 
-$$H=\begin{pmatrix} \omega_1&\omega_2\\  \omega_2&\omega_1\end{pmatrix}$$
-A basis for this system is 
-$$|{0}\rangle=\begin{pmatrix}1\\0  \end{pmatrix}\, ,\quad|{1}\rangle=\begin{pmatrix}0\\1  \end{pmatrix}$$
+3. Find the eigenvalues and eigenvectors of the matrices
 
-Find the eigenvalues and eigenvectors of the Hamiltonian $H$, and express the eigenvectors in terms of $\{|0 \rangle,|1\rangle \}$
+    $$A=\begin{pmatrix} -2&-1&-1\\6&3&2\\0&0&1 \end{pmatrix}\, \quad B=\begin{pmatrix} 1&1&2\\2&2&2\\-1&-1&-1 \end{pmatrix} $$.
 
+4. *The Hadamard gate*
 
-**3)** Find the eigenvalues and eigenvectors of the matrices
+    In one of the problems of the previous section we discussed that an important operator used in quantum computation is the *Hadamard gate*, which is represented by the matrix:
+    $$\hat{H}=\frac{1}{\sqrt{2}}\begin{pmatrix}1&1\\1&-1\end{pmatrix} \, .$$
+    Determine the eigenvalues and eigenvectors of this operator.
 
-$$A=\begin{pmatrix} -2&-1&-1\\6&3&2\\0&0&1 \end{pmatrix}\,  \quad B=\begin{pmatrix} 1&1&2\\2&2&2\\-1&-1&-1 \end{pmatrix} $$.
+5. Show that the Hermitian matrix
 
-**4)** *The Hadamard gate*
+    $$\begin{pmatrix} 0&0&i\\0&1&0\\-i&0&0 \end{pmatrix}$$
+    has only two real eigenvalues and find and orthonormal set of three eigenvectors.
 
-In one of the problems of the previous section we discussed that an important operator used in quantum computation is the *Hadamard gate*, which is represented by the matrix:
-$$\hat{H}=\frac{1}{\sqrt{2}}\begin{pmatrix}1&1\\1&-1\end{pmatrix} \, .$$
-Determine the eigenvalues and eigenvectors of this operator.
+6. Confirm, by explicit calculation, that the eigenvalues of the real, symmetric matrix
 
-**5)** Show that the Hermitian matrix
-
-$$\begin{pmatrix} 0&0&i\\0&1&0\\-i&0&0 \end{pmatrix}$$
-
-has only two real eigenvalues and find and orthonormal set of three eigenvectors.
-
-**6)** 
-Confirm, by explicit calculation, that the eigenvalues of the real, symmetric matrix
-
-$$\begin{pmatrix} 2&1&2\\1&2&2\\2&2&1 \end{pmatrix}$$
-
-are real, and its eigenvectors are orthogonal.
-
-  
-  
+    $$\begin{pmatrix} 2&1&2\\1&2&2\\2&2&1 \end{pmatrix}$$
+    are real, and its eigenvectors are orthogonal.