Transposed matrix . It is said of the matrix obtained by changing the rows for the columns.
Summary
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- 1 Definition
- 2 Properties
- 3 See also
- 4 Sources
Definition
Is A an mxn matrix order, its transposed matrix is defined as the dimension nxm matrix resulting change to the rows of A by the columns of A . The transposed matrix of A is denoted by A T or by A ‘ .
Note that the element in row i and column j of A is the element in row j and column i of A T .
Example: the transposed matrix of the identity matrix is equal to the matrix itself.
Properties
- The transposed matrix of the transposed matrix of Ais A : (A T ) T = A
- The transposition of the sum is the sum of the transpositions: (A + B) T= A T + B T
- Transposition of the product: (A · B) T= B T A T
- A matrix is the same as its transpose if, and only if, it is a symmetric matrix.
- The transpose of a diagonal matrix and square Ais A . Equality is not true if the matrix is diagonal but not square.
- The determinant of a regular matrix is equal to that of its transposition.
