Row Matrix
A row matrix is a matrix that consists of only one row.
The order of a row matrix is , where is the number of columns.
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Its general form is:
Example:
Matrix is a row matrix of order .
A column matrix is a matrix that consists of only one column.
The order of a column matrix is , where is the number of rows.
Its general form is:
Example:
Matrix is a column matrix of order .
A square matrix is a matrix that has the same number of rows and columns.
If , then the matrix is of order .
Its general form is:
In a square matrix, there are:
Main Diagonal (or Principal Diagonal):
The elements (i.e., where ).
Anti-diagonal (or Counter-diagonal):
The elements (i.e., where ).
Example:
Matrix is a square matrix of order . Its main diagonal elements are . Its anti-diagonal elements are .
A rectangular matrix is a matrix where the number of rows and columns are not equal ().
General example:
Matrix above has rows and columns, so its order is . Since the number of rows is not equal to the number of columns (), matrix is a rectangular matrix.
Rectangular matrices with more columns than rows are often called wide, while those with more rows than columns are often called tall.
A wide matrix is a rectangular matrix with more columns than rows ().
Example:
Matrix is a wide matrix of order .
A tall matrix is a rectangular matrix with more rows than columns ().
Example:
Matrix is a tall matrix of order .
A triangular matrix is a square matrix where the elements on one side of the main diagonal are all zero. The diagonal entries themselves may be anything. Upper and lower triangular matrices differ in which side is zero.
An upper triangular matrix is a square matrix where all elements below the main diagonal are zero.
In symbols, whenever .
Example:
A lower triangular matrix is a square matrix where all elements above the main diagonal are zero.
In symbols, whenever .
Example:
A diagonal matrix is a square matrix where all elements outside the main diagonal are zero.
In symbols, whenever . Elements on the main diagonal can be zero or non-zero.
Example:
Matrix is a diagonal matrix of order .
An identity matrix (denoted by or ) is a diagonal matrix where all elements on the main diagonal are .
Example:
The identity matrix is the neutral element of matrix multiplication. If has dimensions , the identity matrix must match the side on which it is multiplied:
A zero matrix (denoted by or ) is a matrix where all elements are zero.
Example:
A symmetric matrix is a square matrix that is equal to its transpose ().
For every pair of indices and , a symmetric matrix satisfies . Entries on opposite sides of the main diagonal are equal.
Example:
In matrix :
These names are not mutually exclusive. An identity matrix is both diagonal and symmetric. However, a nonsquare zero matrix is neither diagonal nor symmetric under the definitions above. Check the shape first, then the pattern of entries.
| Example | Applicable types | Reason |
|---|---|---|
| Identity, diagonal, upper and lower triangular, symmetric | Every off-diagonal entry is zero and every main-diagonal entry is one. | |
| Zero, rectangular, wide | Every entry is zero, but the row and column counts differ. |
Published: . Updated: .