diagonal matrix∗
rspuzio†
2013-03-21 16:12:07
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 × n diagonal matrix is completely
determined by the n entries on the diagonal; all other entries are zero. If the
diagonal entries are a1 , a2 , . . . , an , then we denote the corresponding diagonal
matrix by


a1 0 0 · · · 0
 0 a2 0 · · · 0 




diag(a1 , . . . , an ) =  0 0 a3 · · · 0  .

 ..
..
.. . .

.
.
.
.
0 0 0
an
Examples
1. The identity matrix and zero matrix are diagonal matrices. Also, any 1×1
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. A square matrix is diagonal if and only if it is triangular and normal (see
this page).
∗ hDiagonalMatrixi created: h2013-03-21i by: hrspuzioi version: h34411i Privacy setting:
h1i hDefinitioni h15-00i h15A57i
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3. The eigenvalues of a diagonal matrix A = diag(a1 , . . . , an ) are a1 , . . . , an .
Corresponding eigenvectors are the standard unit vectors in Rn . For the
determinant, we have det A = a1 a2 · · · an , so A is invertible if and only if
all ai are non-zero. Then the inverse is given by
−1
= diag(1/a1 , . . . , 1/an ).
diag(a1 , . . . , an )
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
ediag(a1 ,...,an ) = diag(ea1 , . . . , ean ).
More generally, every analytic function of a diagonal matrix can be computed
entrywise, i.e.:
f (diag(a11 , a22 , ..., ann )) = diag(f (a11 ), f (a22 ), ..., f (ann ))
Remarks
Diagonal matrices are also sometimes called quasi-scalar matrices [?].
References
[1] H. Eves, Elementary Matrix Theory, Dover publications, 1980.
[2] Wikipedia, diagonal matrix.
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