Diagonal Matrix Properties

A (mxm) matrix is a square array of numbers which has equal number of rows and columns here denoted by ‘m’. The elements of a square matrix that start from the left corner towards the right corner like a diagonal is called the Main Diagonal of a Matrix. For example, a 3x3 matrix given by
A=[-1  3  4]
 [2   1  0]
 [7  0   5]
The elements [ -1,1,5] form the elements of the main diagonal. Now consider A=(aij) to be a square matrix then the elements in aii is the diagonal of a matrix.  A square matrix in which all elements except the elements of the leading diagonal are zero is called a Matrix Diagonal. In this all the non diagonal elements are zero. For instance, a square matrix given by A=[aij]nxn to be a diagonal-matrix , aij is equal to zero wherever ‘i’ is not equal to ‘j’. Example of a diagonal-matrix can be given as follows, a 3x3 matrix given by
A=[2    0    0]
     [0   -1    0]
     [0    0    5]

Let us now take a look at the Diagonal Matrix Properties:
• If A and B are two diagonal matrices of the same order then, A+B and AB are also diagonal matrices.  Also AB=BA, that is they commute
• A square matrix is diagonal only if it is triangular and normal
• If A is a diagonal-matrix then the adjugate of A is also a diagonal matrix
• Any diagonal matrix is symmetric
• It is a scalar matrix when all the entries of the diagonal are equal
Determinant Diagonal Matrix
The determinant of a 2x2 square matrix A=[a    b] can be defined as |A|=ad-bc. It is denoted by det(A)
    [c    d]
Let A=[aij] be a nxn matrix where n is greater than or equal to two. We can define determinant,
|A|= a11C11 + a12C12+……..+a1nC1n which is equal to summation(i= 1 to n)[a1iC1i]. If A is a nxn diagonal matrix then |A| is the product of the elements in its main diagonal.
Let A = [a11       0       0……     0]
             [0         a22     0 ……   0]
     [0          0      a33……    0]
     [..         ..      ..     ..]
     [..         ..      ..     ..]
     [0          0        0…. .ann]

be a nxn diagonal matrix, then the determinant of this matrix can be given by a simple formula, det (A)=|A|= pi(i= 1 to n)[aii]
Consider the following example,
A=[2   0    0]
      [0   5    0]
      [0   0    1]

then det(A)=|A|= (2)(5)(1)=10
Note that if any of the elements of the main diagonal is zero then determinant would also be zero.