Begin2.DVI

The transpose matrix

The transpose of a matrix A= (aij)m×nis obtained by interchanging the rows and

columns of the matrix A. The transpose matrix is denote AT= (aji)n×m. That is, if

A=







a 11 a 12... a 1 n

a 21 a 22... a 2 n

..

.

..

. ...

..

.

am 1 am 2... a mn







m×n

then AT=







a 11 a 21... a m 1

a 12 a 22... a m 2

..

.

..

. ...

..

.

a 1 n a 2 n... a mn







n×m

Note that (AT)T =A. If AT =A, then the matrix Ais called a symmetric matrix.

If AT =−A, then Ais called a skew-symmetric matrix. The matrix transpose of a

product satisfies

(AB )T=BTAT, (ABC )T=CTBTAT

so that the transpose of a product is the product of the transposed matrices in reverse

order.

Lower triangular matrices

Matrices which satisfy

A= (aij), where aij = 0 for i < j,

are called lower triangular matrices. Such matrices have zero for elements everywhere

above the main diagonal. Any example of a lower triangular matrix is given in the

figure 10-1.

A=





3 0 0 0

1 2 0 0

4 0 2 0

1 − 1 −2 4





Figure 10-1. A 4 × 4 lower triangular matrix.

Upper triangular matrices

If a square matrix Asatisfies

A= (aij), where aij = 0 for i > j,

it is called an upper triangular matrix. Such matrices have zero for elements every-

where below the main diagonal. An example upper triangular matrix is illustrated

in the figure 10-2.