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# Transpose Matrix

• Date Submitted: 05/30/2011 12:50 AM
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Transpose Matrix
The transpose of one matrix is another matrix that is obtained by using by using rows from the first matrix as columns in the second matrix.
For example,
A =     111 222
333 444
555 666

A' =     111 333 555
222 444 666

Note that the order of a matrix is reversed after it has been transposed. Matrix A is a 3 x 2 matrix, but matrix A' is a 2 x 3 matrix.
Vectors
Vectors are a type of matrix having only one column or one row.
. A matrix with one column (an m × 1 matrix) is called a column vector   and one   row (a 1 × n matrix) is called a row vector.For example, Matrix a is a column vector, and matrix a' is a row vector.
a =     11
12
33

a' =     11 22 33

Square Matrices
A square matrix is an n x n matrix; that is, a matrix with the same number of rows as columns. In this section, we describe several special kinds of square matrix.
• Symmetric matrix. If the transpose of a matrix is equal to itself, that matrix is said to be symmetric. Two examples of symmetric matrices appear below.
A = A' =     1 2
2 3
B = B' =     5 6 7
6 3 2
7 2 1

Diagonal matrix. A diagonal matrix is a special kind of symmetric matrix. It is a         symmetric matrix with zeros in the off-diagonal elements. Two diagonal matrices are shown below.

A =     1 0
0 3
B =     5 0 0
0 3 0
0 0 1

Scalar matrix. A scalar matrix is a special kind of diagonal matrix. It is a diagonal matrix with equal-valued elements along the diagonal. Two examples of a scalar matrix appear below.
A =     3 0
0 3
B =     5 0 0
0 5 0
0 0 5

Matrix Dimension
The number of rows and columns that a matrix has is called its dimension or its order. By convention, rows are listed first; and columns, second. Thus, we would say that the dimension (or order) of the matrix below is 3 x 4, meaning that it has 3 rows and 4 columns.
21 62 33 93
44 95 66 13
77 38 79 33
Matrix...