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Matrix Size & Validation
Vector Algebra
What is Vector?
Vector Norm
Unit Vector
Vector Addition
Vector Subtraction
Vector Scalar Multiple
Vector Multiplication
Vector Inner Product
Vector Outer Product
Vector Cross Product
Vector Triple Cross Product
Vector Triple Dot Product
Scalar Triple Product
Orthogonal & Orthonormal Vector
Cos Angle of Vectors
Scalar and Vector Projection
Matrix Algebra
What is a matrix?
Special Matrices
Matrix One
Null Matrix
Matrix Diagonal Is Diagonal Matrix?
Identity Matrix
Matrix Determinant
Matrix Sum
Matrix Trace
Matrix Basic Operation
Is Equal Matrix?
Matrix Transpose
Matrix Addition
Matrix Subtraction
Matrix Multiplication
Matrix Scalar Multiple
Hadamard Product
Horizontal Concatenation
Vertical Concatenation
Elementary Row Operations
Matrix RREF
Finding inverse using RREF (Gauss-Jordan)
Finding Matrix Rank using RREF
Matrix Inverse
Is Singular Matrix?
Linear Transformation
Matrix Generalized Inverse
Solving System of Linear Equations
Linear combination, Span & Basis Vector
Linearly Dependent & Linearly Independent
Change of basis
Matrix Rank
Matrix Range
Matrix Nullity & Null Space
Eigen System
Matrix Eigen Value & Eigen Vector
Symmetric Matrix
Matrix Eigen Value & Eigen Vector for Symmetric Matrix
Similarity Transformation and Matrix Diagonalization
Matrix Power
Orthogonal Matrix
Spectral Decomposition
Singular Value Decomposition
Resources on Linear Algebra

Matrix Transpose

By Kardi Teknomo, PhD.
LinearAlgebra

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Transpose of a matrix is formed by interchanging the rows and columns of the input matrix.
If the input matrix size is m by n, the transpose matrix size is n by m.


Example:
Matrix Transpose        Matrix Transpose


The notation of matrix transpose varies in the literatures, but in general we have three most used notation of matrix transpose: Matrix TransposeorMatrix Transpose, orMatrix Transpose. In this linear algebra tutorial, I adopt the first notation because it’s clearer in the eyes.

The interactive program below produces matrix transpose of the input matrix. Random Example button provide you will unlimited examples of random matrix. Play with the two buttons or type your own input matrix to gain more understanding about matrix transpose.


Properties

Some important properties of matrix transpose are

  • Transpose of a (column) vector is a row vector and vice versa
  • The transpose of a transposed matrix returns the original matrix,Matrix Transpose
  • Transpose of the summation of two matrices is equal to the summation of their transposes Matrix Transpose
  • Transpose of the product of two matrices is equal to the product of their transposes taken in the reverse order, Matrix Transpose
  • The transpose of the matrix products can be extended to several matrices Matrix Transpose
  • The inverse of a transpose matrix is equal to the transpose of its inverse,Matrix Transpose

See also: matrix addition, matrix multiplication

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This tutorial is copyrighted.

Preferable reference for this tutorial is

Teknomo, Kardi (2011) Linear Algebra tutorial. http:\\people.revoledu.com\kardi\ tutorial\LinearAlgebra\

 

 
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