Diagonal matrices mathematics

Diagonal Matrices

Diagonal matrices are those that have upper and lower triangular matrices. The name “diagonal matrix” means that all the elements that are not on the principal diagonal are zeros. Matrix A = [aij] is said to be diagonal if: 

• A is s square matrix and 
•  aij= 0 when i is greater than or equal to j.

It’s unclear whether a diagonal matrix can have zeros on all or some of its diagonal elements. The only requirement for it to be a diagonal matrix concerns its non-principal diagonal components, which it may or may not satisfy (which must be zeros). Therefore, the diagonal members of a diagonal matrix can either be zeros or non-zeroes.

The properties of diagonal matrices:

This property can be used to our advantage when dealing with diagonal matrices.

• C = AB is a diagonal if A and B are diagonal. C can also be obtained more quickly than a full matrix multiplication using the formula cii=aiibi, with all other entries equal to 0.
• The matrices A and B are diagonal, hence C = AB = BA when multiplying the two.
• The C’s ith row is aii times the B’s ith row, and the ith column is aii times the column (ith) of B if A is a diagonal matrix and B is a general matrix. 

Block Diagonal Matrices: What are they?

The term “block matrix” refers to a matrix that has been divided into sections. Off-diagonal blocks are zero matrices, whereas the main diagonal blocks are square matrices in this sort of square matrix. In this case, there are no non-diagonal blocks. When i is not equal to j and  Dij = 0, the matrix D is called a block diagonal matrix. The following is an example of it:

Inverse Diagonal Matrices

D is a diagonal matrix if the components on the major diagonal of the D are the inverse of the corresponding elements on the main diagonal of the D.

 Let’s consider the diagonal matrices like the following:

 Let’s consider the diagonal matrices like the following:

 Anti-diagonal matrices

It is an anti-diagonal matrix if all entries are zero, except for those on the diagonals from the left corner at the bottom to the upper right corner.

Here’s how anti-diagonal matrices go:

Here’s how anti-diagonal matrices go:

Diagonal Matrices: A Simple Method of Determination

To determine if a matrix has a diagonal, check if it is square and if all the elements other than the major diagonal (the diagonal that goes from top left to bottom right) are. For example, let’s have a look at the following matrices:

Each of the four matrices illustrated above has the following characteristics:

• There is the same number of rows and columns in Matrix A, making it a square matrix (2X2 matrix). Entrances and exits are located on the primary diagonal, respectively. Other items have a value of 0. As a result, we have a matrix where the diagonals are equal.
• There are exactly equal numbers of rows and columns in Matrix B, making it square (matrix). The main diagonal has the numbers 1, 5, and -2. There are no other entries here. As a result, we’re dealing with a diagonal matrix here.
•  If we take a quick look at Matrix C, it might look like a diagonal matrix. As a starting point, however, the matrix is non-square. Therefore, it is not a diagonal matrix.
•  It’s a little difficult to solve this one. A square matrix is the first thing to notice (3X3). You might mistake it for a non-diagonal matrix at first. 

Suppose you examine the definition of a diagonal matrix more closely as per the study material notes on Diagonal matrices. In that case, you will notice that it does not state that the diagonal entries cannot be! The requirement is that all elements other than the diagonal are equal to 0. In other words, it makes no difference if there are diagonal components. Any matrices other than the diagonal are considered diagonal if all of its entries are set to 0.

Matrix D is, in fact, a diagonal matrix, as previously stated!

There are two sorts of diagonal matrices:

• Identity Matrix
• Zero Matrix

There are no entries in the primary diagonal hence this is a square matrix with only 0 entries. 

The identity matrices for 2×2 and 3×3 are presented below.

Identity Matrix

Like the zero matrix, a form of matrix known as the multiplicative identity is commonly referred to as I. As a result, we can write IM = MI = M for any matrix M. Since the identity matrix must be square, unlike the zero matrix, it is sometimes denoted by the notation ‘In‘ for an identity matrix of order ‘n’. For example, M = ImM + MIn = M for an m x n matrix M. Entirely, the other members of the identity matrix are zeros, except for the top-to-bottom right diagonal, which is all one.

For example:

In most cases, an identity matrix can be shown as follows:

Conclusion

We have learned about the basics of Diagonal Matrices and their properties such as C = AB, C = AB = BA, etc. Moreover, we have insight into the block diagonal matrix, where the matrix can be segregated into different sections. We understood Inverse Diagonal Matrices where D is a diagonal matrix if the components on the major diagonal of the D are the inverse of the corresponding elements on the main diagonal of the D. We learned about Anti-diagonal matrices with examples when all entries are zero. Herein, you’ve also understood the simple method of determination for diagonal matrices with examples. Lastly, we learned about the Identity Matrix where M =ImM + MIn = M for an m x n matrix M.