Square matrix

The Square Matrix is a matrix with the same number of rows and columns (m=n). It is referred to as a matrix of order n and denoted by the symbol A(n).

What are the uses of Square Matrices? Square Matrices possess specific characteristics and properties that make them particularly useful in Linear Algebra.

The diagonal of a square matrix

Square matrices are characterized by the presence of two diagonals.

1. The main diagonal. Given a square matrix of order n, the main diagonal is composed of the elements a_i,i for i = 1, ..., n. It is the descending diagonal that starts from the top-left and ends at the bottom-right.
   
2. The secondary diagonal. Given a square matrix of order n, the secondary diagonal is composed of the elements a_i,i for i = n, ..., 1. It is the diagonal that starts from the top-right and ends at the bottom-left.
   

Note. Square matrices are the only matrices that have diagonals. Rectangular matrices, row matrices, column matrices, and null matrices do not have any diagonal.

Diagonal, Upper Triangular, and Lower Triangular Matrices

In certain situations, square matrices can be classified as diagonal, upper triangular, or lower triangular.

• Upper Triangular Matrices. A matrix is said to be upper triangular if a_ij = 0 for every i > j. This means that all elements below the main diagonal are equal to zero.
  
  

  The set of real upper triangular matrices of order n is denoted by U_n(R).

  Strictly Upper Triangular Matrices. A matrix is said to be strictly upper triangular if all elements a_ij are equal to zero for every i ≥ j, which includes the elements on the main diagonal and all elements below it.
  
  A strictly upper triangular matrix is also an upper triangular matrix. The set of real strictly upper triangular matrices of order n is denoted by U*_n(R).

• Lower Triangular Matrices. A matrix is said to be lower triangular if a_ij = 0 for every i < j, meaning all elements above the main diagonal are equal to zero.
  
  

  The set of real lower triangular matrices of order n is denoted by L_n(R).

  Strictly Lower Triangular Matrices. A matrix is said to be strictly lower triangular if all elements aij for every i≤j are equal to zero, including the elements on the main diagonal and all elements above the main diagonal.
  
  A strictly lower triangular matrix is also a lower triangular matrix. The set of real strictly lower triangular matrices of order n is denoted by L*_n(R).

• Diagonal. A square matrix is called diagonal if a_ij = 0 for every i≠j. If all the elements above and below the main diagonal are also zero, then the matrix is said to be diagonal. In fact, a diagonal matrix is both upper triangular and lower triangular.
  
  

  The set of n x n real diagonal matrices is denoted by D_n(R).

Note. The definition of upper or lower triangular does not imply that the other elements must be nonzero.

Special cases of square matrices in linear algebra

The zero matrix is a special case of a square matrix because it is simultaneously both lower and upper triangular, as well as strictly lower and upper triangular, and diagonal.

Another special case is the square matrix of order 1 because it is composed of a single element a1,1.

By definition, a square matrix of order 1 is always lower triangular, upper triangular, and diagonal.

Note. In the case where a_1,1 = 0, the square matrix of order 1 is also strictly lower and upper triangular.