Diagonal Matrix

A matrix is a square braced array of numbers that is separated into rows and columns. When you see a 2×2 matrix, it signifies it contains two rows and two columns. The coefficient of variation, adjoint of a matrix, determinant of a matrix, and inverse of a matrix are all calculated using matrix formulas. The matrix formula comes in handy when we need to compare findings from two distinct surveys that have different values. Matrices come in a variety of sizes, but their shapes are usually the same. The total number of rows and columns in a given matrix is referred to as the matrix’s dimension.

The following is a list of the most popular types of matrices used in linear algebra:

• Column Matrix
• Matrix of Singletons
• Rectangular Matrix 
• Matrix Square
• Matrix of Ones
• Zero Matrix 
• Identity Matrices
• Diagonal Matrix 

These many matrices can be used to categorize data by age group, person, firm, month, and so on. This information can then be used to make judgments and solve a variety of math issues.

• Here’s a rundown of some key aspects to keep in mind while you explore the various forms of matrices:

Row matrices are matrices with only one row and any number of columns.

Column matrices are matrices with one column and any number of rows.

For each given dimension/size of the matrix, constant matrices are matrices in which all of the elements are constants.

• Diagonal Matrix

Let us review a few other sorts of matrices before learning what a diagonal matrix is. Lower triangular matrices and higher triangular matrices are the two varieties of triangular matrices.

• A square matrix with all zeroes above the principal diagonal is called a lower triangular matrix.
• A square matrix with all zeros below the principal diagonal is called an upper triangular matrix.

• What is a Diagonal Matrix, and how does it work?

A diagonal matrix is one that has both upper and bottom triangle elements. The name “diagonal matrix” comes from the fact that all the entries above and below the principal diagonal are zeros. A matrix A = [aij] is considered to be diagonal if it has the following mathematical definition:

• A is a square matrix
• aij 0 when i ≠ j.

Some (or all) of the diagonal elements of a diagonal matrix are zeros. The only criteria for it to be a diagonal matrix is that its non-principal diagonal entries are present (which have to be zeros). In other words, a diagonal matrix’s diagonal elements can be zeros or non-zeroes.

• Anti-Diagonal Matrix

In terms of element placement, an anti-diagonal matrix is just the opposite image of a diagonal matrix. All the entries above and below the diagonal (which is NOT the major diagonal) in an anti-diagonal matrix are zeros. It’s important to remember that any anti-diagonal matrix is also a non-diagonal matrix.

• Properties of a Diagonal Matrix

• Based on its definition, here are the properties of a diagonal matrix.
• Every square matrix is a diagonal matrix.
• A diagonal matrix is one in which the non-principal diagonal members are all zeros, such as the identity matrix, null matrix, or scalar matrix.
• A diagonal matrix is the product of two diagonal matrices.
• The product of two diagonal matrices (of the same order) is a diagonal matrix whose primary diagonal components are the products of the original matrices’ corresponding elements.
• Both addition and multiplication are commutative for diagonal matrices.
• Diagonal matrices are symmetric matrices as for any diagonal matrix A, AT = A.

• Determinant of Diagonal Matrix

The product of a diagonal matrix’s diagonal members is its determinant.

 Thus,

Only if all of a diagonal matrix’s principal diagonal elements are non-zero is it a non-singular matrix (one with a non-zero determinant).

• The inverse of the Diagonal Matrix

The Inverse of a diagonal matrix is a diagonal matrix whose principal diagonal members are the reciprocals of the original matrix’s corresponding elements

As we know the inverse of a matrix A is, A-1 = (adj A) / (det A)

We can see that A-1 is also a diagonal matrix, with its primary diagonal elements being the reciprocals of A’s corresponding elements. Thus,

Conclusion

A matrix is a rectangular array of numbers organized in columns and rows. The numbers are known as the matrix’s entries or elements. Engineering, physics, economics, and statistics, as well as various disciplines of mathematics, all use matrices. A diagonal matrix is a square matrix with zero entries on all off-diagonal sides. (It’s worth noting that a diagonal matrix must be symmetric.) The main diagonal entries may or may not be zero.