Mathematical structure of quantum mechanics

• A quantum description consists of a Hilbert space of states
• Observables are self adjoint operators on the space of states
• Time evolution is given by a one-parameter group of unitary transformations on the Hilbert space of states
• Physical symmetries are realized by unitary transformations

Postulates of quantum mechanics

The mathematical framework can be traced back to the Dirac-von Neumann axioms

Tensor Calculus

• vector (contra-variant vector) - arrow in space
• covector (co-variant vector) - gradient 

Eigenvalues and Eigenvectors

• $\lambda\in \lambda(A) \Leftrightarrow A-\lambda I \text{ is singular} \Leftrightarrow \text{det}(A-\lambda I)=0$
• $\{x\neq0|x\in \mathcal{N}(A-\lambda I)\}$ is the set of all eigenvectors associated with $\lambda$.  
• $\mathcal{N}(A-\lambda I)$ is the eigenspace for A.

Diagonalizability of a matrix

• A nilpotent matrix $A=\{A\in M_n |A^2=0\}$ is not diagonalizable. 
• Two matrices $A$ and $B$ are similar whenever these exists a nonsingular matrix $P$ such that $P^{-1} AP=B$
• A matrix can be diagonalized if it is similar to a diagonal matrix $D$, i.e. $P^{-1}AP=D$
• Or equivalently, $AP_{*,j}=\lambda_j P_{*,j}$
• $A$ is diagonalizable if and only if $A$ possesses a complete set of eigenvectors.
• Or equivalently, the geometric multiplicity of $\lambda_i$ is equal to the algebraic multiplicity of $\lambda_i$ for each $\lambda_i\in \lambda(A)$