7.8. Diagonalization Of A Hermitian Or Skew-Hermitian Matrix
Theorem 7.7.
Every n  n Hermitian or skew-Hermitian matrix A is similar to the diagonal matrix
  diag  1,
, n  of its eigenvalues, i.e.,
  C 1 AC
The matrix A is said to be diagonalizable.
unitary, i.e., C
1
Moreover, the transition matrix C is
C .
†
Proof
Let V be the space Cn of n-tuples of complex numbers.
Let E   e1,
, en  be
the orthonormal basis of unit coordinate vectors. The inner product of any two
n
n
i 1
i 1
elements x   xi ei or x   xi i
n
n
i 1
i 1
and y   yi ei or y   yi i
is
therefore
 x, y  
n
y x
  xi yi
i 1
Let T : V  V be the linear transformation such that mE T   A . By Theorem 7.4,
there exists a basis U   u1,
mU T     diag  1,
, un  of eigenvectors such that
, n 
where k is the eignevalue to which uk belongs.
Since both A and  are matrix
representations of T, they are similar, i.e., there exists a non-singular transition matrix
C   cij  such that   C 1 AC with U  EC , or
n
uk   e j c jk
for all k  1,
,n
j 1
or
 u1,
, un    e1,
, en  C
The last equation shows that the jth column of C consists of components of uj relative
to E. Thus, cij is the ith component of uj so that
u , u  
j
i
n
ui u j   ckj cki   ji
k 1

C tC  I

C 1  C t  C †
QED.
Note
The proof of Theorem 7.7 showed that the transition matrix C that diagonalizes A can
be written as C  U1,
,Un  , where Uk is a column matrix representing the
orthonormal eigenvector uk relative to the unit coordinate basis.
Example 1
The real Hermitian (symmetric) matrix
2 2
A

 2 5
has eigenvalues 1  1 and 2  6 with eigenvectors u1  t  2, 1 and
u2  s 1,2 , respectively. They are orthogoanl and can be normalized by setting
1
. Hence, the diagonalizing matrix is
5
1  2 1
1  2 1

C
C 1  C t 
 1 2 


5
5 1 2 

so that
 1 0
C 1 AC  

 0 6
st
Example 2
If A is already diagonal, then C 1 AC remains diagonal so that only the diagonal
elements are rearranged.