Main diagonal

<h2 id="square-matrices">Square matrices</h2>
For a <a href="/facts/Square_matrix/5TP9mNNn">square matrix</a>, the diagonal (or main diagonal or principal diagonal) is the diagonal line of entries running from the top-left corner to the bottom-right corner.<a class="footnote-ref" id="fnref:1" href="#fn:1">1</a><a class="footnote-ref" id="fnref:2" href="#fn:2">2</a><a class="footnote-ref" id="fnref:3" href="#fn:3">3</a> For a matrix 
 
 
 
 A
 
 
 {\displaystyle A}
 
 with row index specified by 
 
 
 
 i
 
 
 {\displaystyle i}
 
 and column index specified by 
 
 
 
 j
 
 
 {\displaystyle j}
 
, these would be entries 
 
 
 
 
 A
 
 i
 j
 
 
 
 
 {\displaystyle A_{ij}}
 
 with 
 
 
 
 i
 =
 j
 
 
 {\displaystyle i=j}
 
. For example, the <a href="/facts/Identity_matrix/Glbcf6ol">identity matrix</a> can be defined as having entries of 1 on the main diagonal and zeroes elsewhere:

(
            
              
                
                  1
                
                
                  0
                
                
                  0
                
              
              
                
                  0
                
                
                  1
                
                
                  0
                
              
              
                
                  0
                
                
                  0
                
                
                  1
                
              
            
            )
          
        
      
    
    {\displaystyle {\begin{pmatrix}1&0&0\\0&1&0\\0&0&1\end{pmatrix}}}

The <a href="/facts/Trace_(linear_algebra)/dchFFvHt">trace of a matrix</a> is the sum of the diagonal elements. 
The top-right to bottom-left diagonal is sometimes described as the minor diagonal or antidiagonal.
The off-diagonal entries are those not on the main diagonal. A <a href="/facts/Diagonal_matrix/tjIAHrE6">diagonal matrix</a> is one whose off-diagonal entries are all zero.<a class="footnote-ref" id="fnref:4" href="#fn:4">4</a><a class="footnote-ref" id="fnref:5" href="#fn:5">5</a>
A superdiagonal entry is one that is directly above and to the right of the main diagonal.<a class="footnote-ref" id="fnref:6" href="#fn:6">6</a><a class="footnote-ref" id="fnref:7" href="#fn:7">7</a> Just as diagonal entries are those 
 
 
 
 
 A
 
 i
 j
 
 
 
 
 {\displaystyle A_{ij}}
 
 with 
 
 
 
 j
 =
 i
 
 
 {\displaystyle j=i}
 
, the superdiagonal entries are those with 
 
 
 
 j
 =
 i
 +
 1
 
 
 {\displaystyle j=i+1}
 
. For example, the non-zero entries of the following matrix all lie in the superdiagonal:

(
            
              
                
                  0
                
                
                  2
                
                
                  0
                
              
              
                
                  0
                
                
                  0
                
                
                  3
                
              
              
                
                  0
                
                
                  0
                
                
                  0
                
              
            
            )
          
        
      
    
    {\displaystyle {\begin{pmatrix}0&2&0\\0&0&3\\0&0&0\end{pmatrix}}}

Likewise, a subdiagonal entry is one that is directly below and to the left of the main diagonal, that is, an entry 
 
 
 
 
 A
 
 i
 j
 
 
 
 
 {\displaystyle A_{ij}}
 
 with 
 
 
 
 j
 =
 i
 −
 1
 
 
 {\displaystyle j=i-1}
 
.<a class="footnote-ref" id="fnref:8" href="#fn:8">8</a> General matrix diagonals can be specified by an index 
 
 
 
 k
 
 
 {\displaystyle k}
 
 measured relative to the main diagonal: the main diagonal has 
 
 
 
 k
 =
 0
 
 
 {\displaystyle k=0}
 
; the superdiagonal has 
 
 
 
 k
 =
 1
 
 
 {\displaystyle k=1}
 
; the subdiagonal has 
 
 
 
 k
 =
 −
 1
 
 
 {\displaystyle k=-1}
 
; and in general, the 
 
 
 
 k
 
 
 {\displaystyle k}
 
-diagonal consists of the entries 
 
 
 
 
 A
 
 i
 j
 
 
 
 
 {\displaystyle A_{ij}}
 
 with 
 
 
 
 j
 =
 i
 +
 k
 
 
 {\displaystyle j=i+k}
 
.
A <a href="/facts/Band_matrix/Iz803N35">banded matrix</a> is one for which its non-zero elements are restricted to a diagonal band. A <a href="/facts/Tridiagonal_matrix/1SK5A0rP">tridiagonal matrix</a> has only the main diagonal, superdiagonal, and subdiagonal entries as non-zero.

<h2 id="antidiagonal">Antidiagonal</h2>
See also: <a href="/facts/Anti-diagonal_matrix/4U3x8SYQ">Anti-diagonal matrix</a>
The antidiagonal (sometimes counter diagonal, secondary diagonal (*), trailing diagonal, minor diagonal, off diagonal, or bad diagonal) of an order 
 
 
 
 N
 
 
 {\displaystyle N}
 
 <a href="/facts/Square_matrix/5TP9mNNn">square matrix</a> 
 
 
 
 B
 
 
 {\displaystyle B}
 
 is the collection of entries 
 
 
 
 
 b
 
 i
 ,
 j
 
 
 
 
 {\displaystyle b_{i,j}}
 
 such that 
 
 
 
 i
 +
 j
 =
 N
 +
 1
 
 
 {\displaystyle i+j=N+1}
 
 for all 
 
 
 
 1
 ≤
 i
 ,
 j
 ≤
 N
 
 
 {\displaystyle 1\leq i,j\leq N}
 
. That is, it runs from the top right corner to the bottom left corner.

[
            
              
                
                  0
                
                
                  0
                
                
                  
                    
                      1
                    
                  
                
              
              
                
                  0
                
                
                  
                    
                      1
                    
                  
                
                
                  0
                
              
              
                
                  
                    
                      1
                    
                  
                
                
                  0
                
                
                  0
                
              
            
            ]
          
        
      
    
    {\displaystyle {\begin{bmatrix}0&0&\color {red}{1}\\0&\color {red}{1}&0\\\color {red}{1}&0&0\end{bmatrix}}}

(*) Secondary (as well as trailing, minor and off) diagonals very often also mean the (a.k.a. k-th) diagonals parallel to the main or principal diagonals, i.e., 
 
 
 
 
 A
 
 i
 ,
 
 i
 ±
 k
 
 
 
 
 {\displaystyle A_{i,\,i\pm k}}
 
 for some nonzero k =1, 2, 3, ... More generally and universally, the off diagonal elements of a matrix are all elements not on the main diagonal, i.e., with distinct indices i ≠ j.

<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Trace_(matrix)/dchFFvHt">Trace</a></li></ul>
<h2 id="notes">Notes</h2>

<ul><li>Bronson, Richard (1970), Matrix Methods: An Introduction, New York: <a href="/facts/Academic_Press/qxoKjzFy">Academic Press</a>, <a href="/facts/LCCN_(identifier)/5hb5Cw2V">LCCN</a> <a href="https://lccn.loc.gov/70097490">70097490</a></li>
<li>Cullen, Charles G. (1966), Matrices and Linear Transformations, Reading: <a href="/facts/Addison-Wesley/A7UFATEG">Addison-Wesley</a>, <a href="/facts/LCCN_(identifier)/5hb5Cw2V">LCCN</a> <a href="https://lccn.loc.gov/66021267">66021267</a></li>
<li>Herstein, I. N. (1964), Topics In Algebra, Waltham: Blaisdell Publishing Company, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-1114541016 {{citation}}: ISBN / Date incompatibility (help)</li>
<li>Nering, Evar D. (1970), Linear Algebra and Matrix Theory (2nd ed.), New York: <a href="/facts/John_Wiley_%2526_Sons/53g3LqJc">Wiley</a>, <a href="/facts/LCCN_(identifier)/5hb5Cw2V">LCCN</a> <a href="https://lccn.loc.gov/76091646">76091646</a></li>
<li><a href="/facts/Eric_W._Weisstein/FbMuVDep">Weisstein, Eric W.</a> <a href="https://mathworld.wolfram.com/Diagonal.html">"Main diagonal"</a>. <a href="/facts/MathWorld/yc1jodbq">MathWorld</a>.</li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1">Bronson (1970, p. 2) - Bronson, Richard (1970), Matrix Methods: An Introduction, New York: Academic Press, LCCN 70097490 <a href="https://lccn.loc.gov/70097490" target="_blank">https://lccn.loc.gov/70097490</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></li>
<li id="fn:2">Herstein (1964, p. 239) - Herstein, I. N. (1964), Topics In Algebra, Waltham: Blaisdell Publishing Company, ISBN 978-1114541016 <a href="#fnref:2" class="footnote-back-ref">↩</a></li>
<li id="fn:3">Nering (1970, p. 38) - Nering, Evar D. (1970), Linear Algebra and Matrix Theory (2nd ed.), New York: Wiley, LCCN 76091646 <a href="https://lccn.loc.gov/76091646" target="_blank">https://lccn.loc.gov/76091646</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></li>
<li id="fn:4">Herstein (1964, p. 239) - Herstein, I. N. (1964), Topics In Algebra, Waltham: Blaisdell Publishing Company, ISBN 978-1114541016 <a href="#fnref:4" class="footnote-back-ref">↩</a></li>
<li id="fn:5">Nering (1970, p. 38) - Nering, Evar D. (1970), Linear Algebra and Matrix Theory (2nd ed.), New York: Wiley, LCCN 76091646 <a href="https://lccn.loc.gov/76091646" target="_blank">https://lccn.loc.gov/76091646</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></li>
<li id="fn:6">Bronson (1970, pp. 203, 205) - Bronson, Richard (1970), Matrix Methods: An Introduction, New York: Academic Press, LCCN 70097490 <a href="https://lccn.loc.gov/70097490" target="_blank">https://lccn.loc.gov/70097490</a> <a href="#fnref:6" class="footnote-back-ref">↩</a></li>
<li id="fn:7">Herstein (1964, p. 239) - Herstein, I. N. (1964), Topics In Algebra, Waltham: Blaisdell Publishing Company, ISBN 978-1114541016 <a href="#fnref:7" class="footnote-back-ref">↩</a></li>
<li id="fn:8">Cullen (1966, p. 114) - Cullen, Charles G. (1966), Matrices and Linear Transformations, Reading: Addison-Wesley, LCCN 66021267 <a href="https://lccn.loc.gov/66021267" target="_blank">https://lccn.loc.gov/66021267</a> <a href="#fnref:8" class="footnote-back-ref">↩</a></li>
</ol>

Main diagonal open-in-new

Main diagonal