Diagonal matrix

<p>In <a href="/facts/Linear_algebra/8VaB4VWk">linear algebra</a>, a diagonal matrix is a <a href="/facts/Matrix_(mathematics)/qa8FI4ko">matrix</a> in which the entries outside the <a href="/facts/Main_diagonal/rkT2F08Q">main diagonal</a> are all zero; the term usually refers to <a href="/facts/Square_matrices/5TP9mNNn">square matrices</a>. Elements of the main diagonal can either be zero or nonzero. An example of a 2×2 diagonal matrix is 
  
    
      
        
          [
          
            
              
                
                  
                    3
                  
                  
                    0
                  
                
                
                  
                    0
                  
                  
                    2
                  
                
              
            
          
          ]
        
      
    
    {\displaystyle \left[{\begin{smallmatrix}3&0\\0&2\end{smallmatrix}}\right]}
  
, while an example of a 3×3 diagonal matrix is
  
    
      
        
          [
          
            
              
                
                  
                    6
                  
                  
                    0
                  
                  
                    0
                  
                
                
                  
                    0
                  
                  
                    5
                  
                  
                    0
                  
                
                
                  
                    0
                  
                  
                    0
                  
                  
                    4
                  
                
              
            
          
          ]
        
      
    
    {\displaystyle \left[{\begin{smallmatrix}6&0&0\\0&5&0\\0&0&4\end{smallmatrix}}\right]}
  
. An <a href="/facts/Identity_matrix/Glbcf6ol">identity matrix</a> of any size, or any multiple of it is a diagonal matrix called a scalar matrix, for example, 
  
    
      
        
          [
          
            
              
                
                  
                    0.5
                  
                  
                    0
                  
                
                
                  
                    0
                  
                  
                    0.5
                  
                
              
            
          
          ]
        
      
    
    {\displaystyle \left[{\begin{smallmatrix}0.5&0\\0&0.5\end{smallmatrix}}\right]}
  
. 
In <a href="/facts/Geometry/wTQf5ffQ">geometry</a>, a diagonal matrix may be used as a <i><a href="/facts/Scaling_matrix/pFm6fvpu">scaling matrix</a></i>, since matrix multiplication with it results in changing scale (size) and possibly also <a href="/facts/Shape/nww0x85d">shape</a>; only a scalar matrix results in uniform change in scale.
</p>

Diagonal matrix open-in-new

Diagonal matrix