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In linear algebra, matrix transpose is an operator that inverts a matrix diagonally. That is, switch the row and column indexes of matrix A by creating another matrix (among other notations) often called AT.
The matrix transpose was introduced in 1858 by the English mathematician Arthur Cayley. For a logical matrix representing the binary relation R, the transpose corresponds to the inverse relation RT.
Matrix
A matrix, a set of numbers arranged in rows and columns to form an array of rectangles. Numbers are called matrix elements or entries. Matrices have a wide range of uses in various fields of engineering, physics, economics, statistics, and mathematics. Matrix also has important uses in computer graphics that have been used to represent image rotation and other transformations.
Historically, the first thing recognized was not the determinant, but the specific number associated with the arrangement of the squares of numbers called the determinant. The idea of the Matrix as a unit of algebra gradually emerged. The term matrix was introduced by the British mathematician James Sylvester in the 19th century, but it was his friend Arthur Cayley who developed the algebraic aspect of the matrix in two papers in the 1850s. Cayley first applied them to the study of systems of linear equations, but they are still very useful. As Cary recognized, a particular set of matrices forms an algebraic system in which many of the usual arithmetic laws (such as associative and distributive laws) hold, but other laws (such as commutative laws) are invalid. So, these are also important.
Transpose of a Matrix
The new matrix obtained by exchanging the rows and columns of the original matrix is called matrix transpose. If A = [aij] is an m × n matrix, the matrix obtained by swapping the rows and columns of A will be the transposed matrix of A. This is represented by A’ or (AT). That is, if A = [aij] m x n, then A’= [aji] n x m.
for example,
Next is the transpose of A
The transpose of matrix A can be thought of as a matrix formed by rearranging rows into columns and columns into rows. These swaps the indexes of each element. An important property of matrix transpose makes it easy to manipulate the matrix. Also, some important transposed matrices are defined based on their characteristics. If the matrix is equal to its transpose, then the matrix is considered symmetric. If the matrix is equal to the negative number of transposes, the matrix is considered skew symmetric. A matrix conjugate transpose is a matrix transpose in which the elements have been replaced with complex conjugates.
Order of Transpose Matrix
Matrix order represents the number of rows and columns in a particular matrix. All horizontal lines of an element are called rows of a matrix and are indicated by n, and vertical lines of an element are called columns of a matrix and are indicated by m. Together they represent the degree of the matrix written as n × m. And the transposed order of the given matrix is described as m*n.
Transpose of a Square Matrix
The matrix obtained from a particular matrix B after changing rows to columns, columns to rows, and vice versa is called transpose of matrix B. Consider the transpose of a 2×2 and 3×3 square matrix.
Properties of Transpose of a Matrix
Transpose of the Transpose Matrix: The transpose of a matrix is the original matrix itself. It’s clear to understand that exchanging rows and columns twice produces the original matrix itself. Mathematically, you can write
(A`) `= A.
Addition Properties of Matrix: Adding a matrix transpose is the same as transposing the result of adding the original matrix. Mathematically, you can write:
(A + B) `= A` + B`
Multiplication by Constant: Transposing a matrix by a constant is always the same as multiplying that constant by the transpose of the original matrix. Mathematically, this property can be written as:
kA` = (kA) `
Multiplication Property of Transpose: A transpose that multiplies two matrices is the same as multiplying the transposes of each matrix in reverse order. The matrix transpose multiplication property can be written as:
(AB) `= B`A`
CONCLUSION
Transposing a matrix in linear algebra is one of the most widely used matrix transformation methods. A matrix is a collection of rectangular arrays of numbers or functions arranged in a specified number of rows and columns. The numbers or functions stored in a matrix are called matrix elements or entries. Matrix is shown in capital letters in square brackets and can be represented as a vertical matrix, a horizontal matrix, and a square matrix. Matrix transpose of a particular matrix is obtained by exchanging rows for columns or columns for rows.
