Sylvester's law of inertia

<p>Sylvester's law of inertia is a <a href="/facts/Theorem/f2Pe1FMD">theorem</a> in <a href="/facts/Matrix_algebra/oj5HBr8B">matrix algebra</a> about certain properties of the <a href="/facts/Gram_matrix/SGb2VGTU">coefficient matrix</a> of a <a href="/facts/Real_number/R02gw5Pb">real</a> <a href="/facts/Quadratic_form/l2n1jXPe">quadratic form</a> that remain <a href="/facts/Invariant_(mathematics)/9TeNN2ed">invariant</a> under a <a href="/facts/Change_of_basis/Hnu1Spcc">change of basis</a>. Namely, if 
  
    
      
        A
      
    
    {\displaystyle A}
  
 is a <a href="/facts/Symmetric_matrix/08CGwjNl">symmetric matrix</a>, then for any invertible matrix 
  
    
      
        S
      
    
    {\displaystyle S}
  
, the number of positive, negative and zero eigenvalues (called the inertia of the matrix) of 
  
    
      
        D
        =
        S
        A
        
          S
          
            
              T
            
          
        
      
    
    {\displaystyle D=SAS^{\mathrm {T} }}
  
 is constant. This result is particularly useful when 
  
    
      
        D
      
    
    {\displaystyle D}
  
 is diagonal, as the inertia of a diagonal matrix can easily be obtained by looking at the sign of its diagonal elements.
</p><p>This property is named after <a href="/facts/James_Joseph_Sylvester/6NNurKJH">James Joseph Sylvester</a> who published its proof in 1852.
</p>

Sylvester's law of inertia open-in-new

Sylvester's law of inertia