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Singular and non-singular matrices form the basis of linear algebra. The term “matrix” is a Latin word meaning “wipe the clean slate.” It is an array of numbers (aka coefficients) that can be transposed in many ways and multiplied by other matrices to produce the resulting matrix. When the number of rows or columns is restricted, the matrix is said to be “singular,” and the other one is “non-singular.”
A matrix X is called “singular” if and only if X = INFINITI(X), i.e., X has all its nonzero elements equal to zero. Else, it will be called “non-singular.”
What is a Matrix?
A matrix can be defined as a rectangular array of numbers.
I.E., Each row consists of a list of numbers in sorted order, each column consists of a list of numbers in sorted order, and the sum of all the entries in an element is zero.
*The total size of all entries should be 1 or more; the rest are 0 (zero).
Singular and Non-singular Matrix:
Singular:
Singular matrix, also known as singular value decomposition, SVD, or eigenvalue decomposition (EVD),
is a technique for analysing the properties of large square matrices. Its main application is to solve the systems of linear equations that arise in linear algebra and other mathematical subjects. Several methods for decomposing a square matrix are used, but singular value decomposition is probably the most widely used. The SVD can be reformulated as a diagonalization problem: finding a basis for the large matrix. It acts as if its coefficients are distributed along with an orthonormal basis.
Non-singular:
A non-singular matrix is one in which “all” the entries are not zero. It is more specifically a square matrix in which all its elements above the main diagonal are nonzero.
It can be shown that if X = INFINITI(X), then.
X = UDV^{-1} where U and V are orthogonal matrices. In this case, X is singular since its rows and columns cannot both be orthonormal.
The determinant of X = INFINITI(X) is zero. This will be a necessary and sufficient condition for A to be singular below. The determinant is defined as the product of the diagonal elements,
In particular, if some row or column has all its entries equal to zero, then its determinant is zero. In that case, the whole matrix is called singular.
If X is a non-singular matrix (that is, if none of its entries are zero), then the determinant of X will not be zero. In this case, the diagonal elements of X are all different, and thus 1·X·X = (1 + 1 + 1)·(1 + 1 + 1) = 1, with no zero diagonal entries (and hence no zero off-diagonal entries).
Singular and non-singular matrices have similar properties in linear algebra but with some important distinctions. The distinction is between the existence or nonexistence of certain properties, especially regarding the solution to linear equations. For instance, every singular matrix can be written as a product of a matrix inversion plus a transpose. In contrast, the fact that a non-singular matrix can be written as such a product is not true without constraints, i.e., if “X” is such a transpose in the product, one of its columns (or rows) must be a vector.
Singular and Non-singular Matrix Example:
Singular Matrix Example
Determinant of Matrix = ( 1*2 – 1*2 ) = 0
Hence, it is a singular matrix.
Non-Singular Matrix Example:
Determinants of matrix: ( 3*2 – 1*2 ) = 4 = Nonzero
Hence, it is a non-singular matrix
Difference between singular and non-singular matrix:
The following table shows the difference between singular and non-singular matrix:
Conclusion:
A matrix can be of two types, i.e., the Singular and non-singular matrix. If all the numbers it has are zero on its main diagonal, then the matrix is said to be zero or singular and cannot be used for computation. It is not allowable to perform operations with zero or singular matrices. A matrix can have entries that are not zero in both its main diagonal and its sub-diagonal. This type of matrix is called a non-singular or “non zero matrix.” As a result, it is possible to compute both the determinant and the inverse of any nonzero matrix.
