Linear-algebra

matrices tensor-product

Vector space

A vector space consists of a set of vectors $\{|\alpha ⟩,|\beta ⟩,|\gamma ⟩,...\}$ and scalars $\{a,b,c,...\}$

• linear combination = $a|\alpha ⟩+b|\beta ⟩+c|\gamma ⟩+...$
• A vector $|\lambda ⟩$ is said to be linearly independent if it cannot be written as a linear combination of the set of vectors
  • the unit vector $\hat{k}$ is linearly independent to the unit vectors $\hat{i}$ and $\hat{j}$, but any vector in the xy plane would be a linearly dependent on $\hat{i}$ and $\hat{j}$
• a collection of vectors span the space when every vector can be written as a linear combination of the set of vectors
• basis = a collection of linearly independent vectors that span the space
• dimension of the vector space = # of bases

inner product

The dot product of an n-dimensional vector space is called the inner product

norm = $||\alpha ||=\sqrt{⟨\alpha |\alpha ⟩}$

• normalized = $||\alpha ||=1$
• orthogonal = $⟨\alpha |\beta ⟩=0$
• orthonormal set = a collection of orthogonal normalized vectors $⟨{\alpha }_i|{\alpha }_j⟩={\delta }_{ij}$ where
  • when $i=j\to {\delta }_{ij}=1$
  • when $i\ne j\to {\delta }_{ij}=0$

Schwartz inequality $|⟨\alpha |\beta ⟩{|}^2\le ⟨\alpha |\alpha ⟩⟨\beta |\beta ⟩$ angle between two vectors $\cos \theta =\sqrt{\frac{⟨\alpha |\beta ⟩⟨\beta |\alpha ⟩}{||\alpha ||||\beta ||}}$

Gram-Schmidt Procedure

Given a set of linearly independent vectors $\{v_1,v_2,v_3,\dots \}$, the Gram-Schmidt produces an orthogonal basis $\{u_1,u_2,u_3,\dots \}$, followed by an orthonormal basis $\{e_1,e_2,e_3,\dots \}$

\hat{e_{1}} = \frac{v_{1}}{\sqrt{ v_{1} \cdot v_{1} }} \\ \text{For } e_{2} : \vec{e_{2}} = v_{2} - \hat{e_{1}} (\hat{e_{1}} \cdot v_{2}) \\ \hat{e_{2}} = \frac{e_{2}}{\sqrt{ e_{2} \cdot e_{2} }} \\ \text{For } e_{3} : \vec{e_{3}} =v_{3} - \hat{e_{1}} (\hat{e_{1}} \cdot v_{3} ) - \hat{e_{2}} (\hat{e_{2}} \cdot v_{3}) \\ \hat{e_{3}} = \frac{e_{3}}{\sqrt{ e_{3} \cdot e_{3} }} \end{align}$$ So, general formula = $$\vec{e_{n}} = v_{n} - \sum_{i<n} \hat{e_{i}} (\hat{e_{i} \cdot v_{n}}) , \hat{e_{n}} = \frac{\vec{e_{n}}}{\sqrt{ e_{n} \cdot e_{n} }}$$