---
title: "Cross Gramian"
chunk: 1/1
source: "https://en.wikipedia.org/wiki/Cross_Gramian"
category: "reference"
tags: "science, encyclopedia"
date_saved: "2026-05-05T12:27:58.997582+00:00"
instance: "kb-cron"
---

In control theory, the cross Gramian (
  
    
      
        
          W
          
            X
          
        
      
    
    {\displaystyle W_{X}}
  
, also referred to by 
  
    
      
        
          W
          
            C
            O
          
        
      
    
    {\displaystyle W_{CO}}
  
) is a Gramian matrix used to determine how controllable and observable a linear system is.
For the stable time-invariant linear system

  
    
      
        
          
            
              x
              ˙
            
          
        
        =
        A
        x
        +
        B
        u
        
      
    
    {\displaystyle {\dot {x}}=Ax+Bu\,}
  

  
    
      
        y
        =
        C
        x
        
      
    
    {\displaystyle y=Cx\,}
  

the cross Gramian is defined as:

  
    
      
        
          W
          
            X
          
        
        :=
        
          ∫
          
            0
          
          
            ∞
          
        
        
          e
          
            A
            t
          
        
        B
        C
        
          e
          
            A
            t
          
        
        d
        t
        
      
    
    {\displaystyle W_{X}:=\int _{0}^{\infty }e^{At}BCe^{At}dt\,}
  

and thus also given by the solution to  the Sylvester equation:

  
    
      
        A
        
          W
          
            X
          
        
        +
        
          W
          
            X
          
        
        A
        =
        −
        B
        C
        
      
    
    {\displaystyle AW_{X}+W_{X}A=-BC\,}
  

This means the cross Gramian is not strictly a Gramian matrix, since it is generally neither positive semi-definite nor symmetric.
The triple 
  
    
      
        (
        A
        ,
        B
        ,
        C
        )
      
    
    {\displaystyle (A,B,C)}
  
 is controllable and observable, and hence minimal, if and only if the matrix 
  
    
      
        
          W
          
            X
          
        
      
    
    {\displaystyle W_{X}}
  
 is nonsingular, (i.e. 
  
    
      
        
          W
          
            X
          
        
      
    
    {\displaystyle W_{X}}
  
 has full rank, for any 
  
    
      
        t
        >
        0
      
    
    {\displaystyle t>0}
  
).
If the associated system 
  
    
      
        (
        A
        ,
        B
        ,
        C
        )
      
    
    {\displaystyle (A,B,C)}
  
 is furthermore symmetric, such that there exists a transformation 
  
    
      
        J
      
    
    {\displaystyle J}
  
 with

  
    
      
        A
        J
        =
        J
        
          A
          
            T
          
        
        
      
    
    {\displaystyle AJ=JA^{T}\,}
  

  
    
      
        B
        =
        J
        
          C
          
            T
          
        
        
      
    
    {\displaystyle B=JC^{T}\,}
  

then the absolute value of the eigenvalues of the cross Gramian equal Hankel singular values:

  
    
      
        
          |
        
        λ
        (
        
          W
          
            X
          
        
        )
        
          |
        
        =
        
          
            λ
            (
            
              W
              
                C
              
            
            
              W
              
                O
              
            
            )
          
        
        .
        
      
    
    {\displaystyle |\lambda (W_{X})|={\sqrt {\lambda (W_{C}W_{O})}}.\,}
  

Thus the direct truncation of the Eigendecomposition of the cross Gramian allows model order reduction (see [1]) without a balancing procedure as opposed to balanced truncation.
The cross Gramian has also applications in decentralized control, sensitivity analysis, and the inverse scattering transform.


== See also ==
Controllability Gramian
Observability Gramian


== References ==