# Diagonal Matrix

A Diagonal Matrix is a matrix in which the entries outside the main diagonal or all zero and in which some entries in the main diagonal are not zero.

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**Example(s):**- an Identity Matrix.

**See:** Matrix Main Diagonal.

## References

### 2011

- http://en.wikipedia.org/wiki/Diagonal_matrix
- In linear algebra, a
**diagonal matrix**is a matrix (usually a square matrix) in which the entries outside the main diagonal (↘) are all zero. The diagonal entries themselves may or may not be zero. Thus, the matrix D = (d_{i,j}) with n columns and n rows is diagonal if:[math]\displaystyle{ d_{i,j} = 0 \mbox{ if } i \ne j \qquad \forall i,j \in \{1, 2, \ldots, n\}. }[/math]For example, the following matrix is diagonal: :[math]\displaystyle{ \begin{bmatrix} 1 & 0 & 0\\ 0 & 4 & 0\\ 0 & 0 & -3\end{bmatrix} }[/math]

The term

*diagonal matrix*may sometimes refer to a rectangular diagonal matrix, which is an*m*-by-*n*matrix with only the entries of the form d_{i,i}possibly non-zero.For example: [math]\displaystyle{ \begin{bmatrix} 1 & 0 & 0\\ 0 & 4 & 0\\ 0 & 0 & -3\\ 0 & 0 & 0\\ \end{bmatrix} }[/math] or [math]\displaystyle{ \begin{bmatrix} 1 & 0 & 0 & 0 & 0\\ 0 & 4 & 0& 0 & 0\\ 0 & 0 & -3& 0 & 0\end{bmatrix} }[/math]

However, in the remainder of this article we will consider only square matrices. Any square diagonal matrix is also a symmetric matrix. Also, if the entries come from the field

**R**or**C**, then it is a normal matrix as well. Equivalently, we can define a diagonal matrix as a matrix that is both upper- and lower-triangular. The identity matrix*I*_{n}and any square zero matrix are diagonal. A one-dimensional matrix is always diagonal.

- In linear algebra, a