"Those who stoop to conquer are fools."

- Ezslax

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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