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Homediagonal matrix

## Primary tabs

# diagonal matrix

Definition
Let $A$ be a square matrix (with entries in any field).
If all off-diagonal entries of $A$ are zero, then $A$ is a
*diagonal matrix*.

From the definition, we see that an $n\times n$ diagonal matrix is completely determined by the $n$ entries on the diagonal; all other entries are zero. If the diagonal entries are $a_{1},a_{2},\ldots,a_{n}$, then we denote the corresponding diagonal matrix by

$\operatorname{diag}(a_{1},\ldots,a_{n})=\begin{pmatrix}a_{{1}}&0&0&\cdots&0\\ 0&a_{{2}}&0&\cdots&0\\ 0&0&a_{{3}}&\cdots&0\\ \vdots&\vdots&\vdots&\ddots&\\ 0&0&0&&a_{{n}}\end{pmatrix}.$ |

# Examples

1. The identity matrix and zero matrix are diagonal matrices. Also, any $1\times 1$ matrix is a diagonal matrix.

2. A matrix $A$ is a diagonal matrix if and only if $A$ is both an upper and lower triangular matrix.

# Properties

1. If $A$ and $B$ are diagonal matrices of same order, then $A+B$ and $AB$ are again a diagonal matrix. Further, diagonal matrices commute, i.e., $AB=BA$. It follows that real (and complex) diagonal matrices are normal matrices.

2. 3. The eigenvalues of a diagonal matrix $A=\operatorname{diag}(a_{1},\ldots,a_{n})$ are $a_{1},\ldots,a_{n}$. Corresponding eigenvectors are the standard unit vectors in $\mathbb{R}^{n}$. For the determinant, we have $\det A=a_{1}a_{2}\cdots a_{n}$, so $A$ is invertible if and only if all $a_{i}$ are non-zero. Then the inverse is given by

$\big(\operatorname{diag}(a_{1},\ldots,a_{n})\big)^{{-1}}=\operatorname{diag}(1% /a_{1},\ldots,1/a_{n}).$ 4. If $A$ is a diagonal matrix, then the adjugate of $A$ is also a diagonal matrix.

5. The matrix exponential of a diagonal matrix is

$e^{{\operatorname{diag}(a_{1},\ldots,a_{n})}}=\operatorname{diag}(e^{{a_{1}}},% \ldots,e^{{a_{n}}}).$

More generally, every analytic function of a diagonal matrix can be computed entrywise, i.e.:

$f(\operatorname{diag}(a_{{11}},a_{{22}},...,a_{{nn}}))=\operatorname{diag}(f(a% _{{11}}),f(a_{{22}}),...,f(a_{{nn}}))$ |

# Remarks

# References

- 1
H. Eves,
*Elementary Matrix Theory*, Dover publications, 1980. - 2 Wikipedia, diagonal matrix.

## Mathematics Subject Classification

15-00*no label found*15A57

*no label found*

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