Replacement of rows by columns in a matrix of dimension
gives the so-called
In particular, for a vector-row the transposed matrix is the vector-column
The basic properties of the transposed matrix:
2) the transposed matrix of the sum of matrices is equal to the sum of the transposed matrix addends, that is
3) the transposed matrix of the product of matrices is equal to the product of the transposed matrix factors, taken upside-down:
For a square matrix the obvious equality takes place:
If the matrix coincides with the transposed one, that is
then it is called
Apparently, that the product is a symmetric matrix as, using property 3, we’ll receive:
E x a m p l e . The matrix
Calculate the products
and
S o l u t i o n .
As one would expect, the symmetric matrices have been received. |

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>> Applied Mathematics
>> Matrix Algebra
>> Principles of Matrix Calculation
>> The transposed matrix