Some Special Types of Matrices
A matrix is if it has rows and columns.[more]
If there is only one row or column, the matrix can be treated like a vector.
For a square matrix, the number of rows equals the number of columns.
A zero matrix acts like the number zero for matrices of the same dimensions.
The main diagonal of a square matrix runs from the top-left corner to the bottom-left corner.
The identity matrix is square, with ones on the main diagonal and zeros elsewhere. It acts like the number one for matrix multiplication.
A diagonal matrix is a square matrix that has zeros off the main diagonal.
Let be and , where and .
The transpose of the matrix , written , reverses the rows and columns of , so that is and . Another way to think of is as the reflection of in its main diagonal.
Let be an square matrix and , where .
A square matrix is symmetric if , so that .
A square matrix is skew-symmetric if , so that . A skew-symmetric matrix is therefore zero on the main diagonal.
A square matrix is upper triangular if it is zero below the main diagonal.
A square matrix is lower triangular if it is zero above the main diagonal.[less]