| Version current |
Version 8 |
| {\bf Definition} |
{\bf Definition} |
| Let $A$ be a square matrix (with entries in any field). |
Let $A$ be a square matrix (with entries in any field). |
| If all off-diagonal entries of $A$ are zero, then $A$ is a |
If all off-diagonal entries of $A$ are zero, then $A$ is a |
| \emph{diagonal matrix}. |
\emph{diagonal matrix}. |
|
|
| From the definition, we see that an $n\times n$ diagonal matrix is |
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 |
completely determined by the $n$ entries on the diagonal; all other entries |
| are zero. If the diagonal entries are $a_1, a_2, \ldots, a_n$, |
are zero. If the diagonal entries are $a_1, a_2, \ldots, a_n$, |
| then we denote the corresponding diagonal matrix by |
then we denote the corresponding diagonal matrix by |
| $$ \diag(a_1,\ldots, a_n) = \begin{pmatrix} |
$$ \diag(a_1,\ldots, a_n) = \begin{pmatrix} |
| a_{1} & 0 & 0 & \cdots & 0 \\ |
a_{1} & 0 & 0 & \cdots & 0 \\ |
| 0 & a_{2} & 0 & \cdots & 0 \\ |
0 & a_{2} & 0 & \cdots & 0 \\ |
| 0 & 0 & a_{3} & \cdots & 0 \\ |
0 & 0 & a_{3} & \cdots & 0 \\ |
| \vdots & \vdots & \vdots & \ddots & \\ |
\vdots & \vdots & \vdots & \ddots & \\ |
| 0 & 0 & 0 & & a_{n} |
0 & 0 & 0 & & a_{n} |
| \end{pmatrix}. $$ |
\end{pmatrix}. $$ |
|
|
| \subsubsection*{Examples} |
\subsubsection*{Examples} |
| \begin{enumerate} |
\begin{enumerate} |
| \item The identity matrix and zero matrix are diagonal matrices. Also, |
\item The identity matrix and zero matrix are diagonal matrices. Also, |
| any $1\times 1$ matrix is a diagonal matrix. |
any $1\times 1$ matrix is a diagonal matrix. |
| \item A matrix $A$ is a diagonal matrix if and only if $A$ is |
\item A matrix $A$ is a diagonal matrix if and only if $A$ is |
| both an upper and lower triangular matrix. |
both an upper and lower triangular matrix. |
| \end{enumerate} |
\end{enumerate} |
|
|
| \subsubsection*{Properties} |
\subsubsection*{Properties} |
| \begin{enumerate} |
\begin{enumerate} |
| \item If $A$ and $B$ are diagonal matrices of same order, then |
\item If $A$ and $B$ are diagonal matrices of same order, then |
| $A+B$ and $AB$ are again a diagonal matrix. Further, diagonal matrices |
$A+B$ and $AB$ are again a diagonal matrix. Further, diagonal matrices |
| commute, i.e., $AB=BA$. It follows that real (and complex) |
commute, i.e., $AB=BA$. It follows that real (and complex) |
| diagonal matrices are normal matrices. |
diagonal matrices are normal matrices. |
| \item A square matrix is diagonal if and only if it is |
\item A square matrix is diagonal if and only if it is |
| triangular and normal (see \PMlinkname{this page}{TheoremForNormalTriangularMatrices}). |
triangular and normal (see \PMlinkname{this page}{TheoremForNormalTriangularMatrices}). |
| \item The eigenvalues of a diagonal matrix |
\item The eigenvalues of a diagonal matrix |
| $A=\diag(a_1,\ldots, a_n)$ are $a_1, \ldots, a_n$. |
$A=\diag(a_1,\ldots, a_n)$ are $a_1, \ldots, a_n$. |
| Corresponding eigenvectors are the standard unit vectors in $\sR^n$. |
Corresponding eigenvectors are the standard unit vectors in $\sR^n$. |
| For the determinant, we have $\det A = a_1 a_2 \cdots a_n$, so |
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. |
$A$ is invertible if and only if all $a_i$ are non-zero. |
| Then the inverse is given by |
Then the inverse is given by |
| $$ |
$$ |
| \big( \diag(a_1,\ldots, a_n)\big)^{-1} = \diag(1/a_1, \ldots, 1/a_n). |
\big( \diag(a_1,\ldots, a_n)\big)^{-1} = \diag(1/a_1, \ldots, 1/a_n). |
| $$ |
$$ |
| \item If $A$ is a diagonal matrix, then the adjugate of $A$ is also a diagonal matrix. |
\item If $A$ is a diagonal matrix, then the adjugate of $A$ is also a diagonal matrix. |
| \item The matrix exponential of a diagonal matrix is |
\item The matrix exponential of a diagonal matrix is |
| $$ |
$$ |
| e^{\diag(a_1,\ldots, a_n)} = \diag(e^{a_1}, \ldots, e^{a_n}). |
e^{\diag(a_1,\ldots, a_n)} = \diag(e^{a_1}, \ldots, e^{a_n}). |
| $$ |
$$ |
| \end{enumerate} |
\end{enumerate} |
|
More generally, every analytic function of a diagonal matrix can be computed entrywise, i.e.:
|
More generally, every analytic function of a diagonal matrix is entrywise, i.e.:
|
|
\[ f(\diag(a_{11},a_{22},...,a_{nn}))= \diag(f(a_{11}),f(a_{22}),...,f(a_{nn})) \]
|
\[ f(diag(a_{11},a_{22},...,a_{nn}))= diag(f(a_{11}),f(a_{22}),...,f(a_{nn})) \]
|
|
|
| \subsubsection*{Remarks} |
\subsubsection*{Remarks} |
| Diagonal matrices are also sometimes called \emph{quasi-scalar matrices} \cite{eves}. |
Diagonal matrices are also sometimes called \emph{quasi-scalar matrices} \cite{eves}. |
|
|
| \begin{thebibliography}{9} |
\begin{thebibliography}{9} |
| \bibitem {eves} H. Eves, |
\bibitem {eves} H. Eves, |
| \emph{Elementary Matrix Theory}, |
\emph{Elementary Matrix Theory}, |
| Dover publications, 1980. |
Dover publications, 1980. |
| \bibitem{wiki_diagonal} Wikipedia, |
\bibitem{wiki_diagonal} Wikipedia, |
| \PMlinkexternal{diagonal matrix}{http://www.wikipedia.org/wiki/Diagonal_matrix}. |
\PMlinkexternal{diagonal matrix}{http://www.wikipedia.org/wiki/Diagonal_matrix}. |
| \end{thebibliography} |
\end{thebibliography} |
