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...
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