page_title: Characteristic mode analysis
section_title: Matrix formulation
hierarchical_section_title: Characteristic mode analysis / Matrix formulation
caption_attribution_description: English: The first (dominant) characteristic mode.
context_page_description: Characteristic modes form a set of functions which, under specific boundary conditions, diagonalizes operator relating field and induced sources. Under certain conditions, the set of the CM is unique and complete and thereby capable of describing the behavior of a studied object in full.
context_section_description: Discretization  of the body of the scatterer  into  subdomains as  and using a set of linearly independent piece-wise continuous functions , , allows current density  to be represented as
  and by applying the Galerkin method, the impedance operator (2)
  The eigenvalue problem (1) is then recast into its matrix form
  which can easily be solved using, e.g., the generalized Schur decomposition or the implicitly restarted Arnoldi method yielding a finite set of expansion coefficients  and associated characteristic numbers . The properties of the CM decomposition are investigated below.
caption: The first (dominant) characteristic mode of a shape 
    
      
        
          
            Ω
            
              M
            
          
        
      
      {\displaystyle \Omega ^{M}}
    
  .
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