# User:Margav06/sandbox/Click here to continue/Notation

**Notations**

In this document, matrices are denoted by boldface letters (e.g.,), column matrices are denoted by lowercase boldface letters (e.g.,), scalars are denoted by simple letters (e.g.,γ ), and operators are denoted by script letters (e.g., : →. The set of by real matrices is denoted as , the set of by complex matrices is denoted as , and the set of symmetric matrices is denoted as . The identity matrix is written as **1** and a matrix filled with zeros is written as **0**. The dimensions of **1** and **0** are specified when necessary. Repeated blocks within symmetric matrices are replaced by **∗** for brevity and clarity. The conjugate transpose or Hermitian transpose of the matrix is denoted by . The notation He{·} is used as a shorthand in situations with limited space, where He{·} = (·) + (·) . The real and imaginary parts of the complex number are denoted as Re() and Im(), respectively. The Kroenecker product of two matrices is denoted by ⊗.

Consider the square matrix . The eigenvalues of are denoted by . The matrix A is Hurwitz if all of its eigenvalues are in the open left-half complex plane
(i.e., Re ). A matrix is Schur if all of its eigenvalues are strictly within a unit disk centered at the origin of the complex plane (i.e., . If , then the minimum eigenvalue of **A** is denoted by and its maximum eigenvalue is denoted by .

Consider the matrix **B** . The minimum singular value of **B** is denoted by (**B**) and its maximum singular value is denoted by (**B**). The range and nullspace of B are denoted by (**B**)
and (**B**), respectively. The Frobenius norm of **B** is ||**B**|| = .

A state-space realization of the continuous-time linear time-invariant (LTI) system

,

.

is often written compactly as (**A, B,C,D**) in this document. The argument of time is often omitted
in continuous-time state-space realizations, unless needed to prevent ambiguity.
A state-space realization of the discrete-time LTI system

is often written compactly as .

The ∞ norm of the LTI system is denoted by ||||∞ and the norm of is denoted by
||||.

The inner product spaces for continuous-time signals are defined as follows.

The inner product sequence spaces ℓ2 and ℓ2e for discrete-time signals are defined as follows.