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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...

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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