 Some Special Types of Matrices

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A matrix is if it has rows and columns.

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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.

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Contributed by: George Beck  (July 2018)
Open content licensed under CC BY-NC-SA

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George Beck

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