diagonal matrix

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

From the definition, we see that an $n\times n$ diagonal matrix iscompletely determined by the $n$ entries on the diagonal; all other entriesare zero. If the diagonal entries are $a_1,a_2,\mathrm{\dots },a_n$,then we denote the corresponding diagonal matrix by

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 $A$ isboth 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 matricescommute, i.e., $AB=BA$. It follows that real (and complex)diagonal matrices are normal matrices.

2. A square matrix is diagonal if and only if it istriangular and normal (see this page (http://planetmath.org/TheoremForNormalTriangularMatrices)).

3. The eigenvalues of a diagonal matrix$A=\mathrm{diag}(a_1,\mathrm{\dots },a_n)$ are $a_1,\mathrm{\dots },a_n$.Corresponding eigenvectors are the standard unit vectors in ${ℝ}^n$.For the determinant, we have $detA=a_1{a}_2\mathrm{\cdots }{a}_n$, so$A$ is invertible if and only if all $a_i$ are non-zero.Then the inverse is given by

   $${\left(\mathrm{diag}(a_1,\mathrm{\dots },a_n)\right)}^{-1}=\mathrm{diag}(1/a_1,\mathrm{\dots },1/a_n).$$

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

   $$e^{\mathrm{diag}(a_1,\mathrm{\dots },a_n)}=\mathrm{diag}(e^{a_1},\mathrm{\dots },e^{a_n}).$$

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

$$f(\mathrm{diag}(a_{11},a_{22},\mathrm{\dots },a_{nn}))=\mathrm{diag}(f(a_{11}),f(a_{22}),\mathrm{\dots },f(a_{nn}))$$

Remarks

Diagonal matrices are also sometimes called quasi-scalar matrices [1].