This paper provides a constructive method using unitary diagonalizable elements to obtain all hermitian matrices
A
A
in
M
n
(
C
)
M_n(\Bbb C)
such that
‖
A
‖
=
min
B
∈
B
‖
A
+
B
‖
,
\begin{equation*} \|A\|=\min _{B\in \mathcal {B}}\|A+B\|, \end{equation*}
where
B
\mathcal {B}
is a C*-subalgebra of
M
n
(
C
)
M_n(\Bbb C)
,
‖
⋅
‖
\|\cdot \|
denotes the operator norm. Such an
A
A
is called
B
\mathcal {B}
-minimal. Moreover, for a C*-subalgebra
B
\mathcal {B}
determined by a conditional expectation from
M
n
(
C
)
M_n(\Bbb C)
onto it, this paper constructs
⨁
i
=
1
k
B
\bigoplus _{i=1}^k\mathcal {B}
-minimal hermitian matrices in
M
k
n
(
C
)
M_{kn}(\Bbb C)
through
B
\mathcal {B}
-minimal hermitian matrices in
M
n
(
C
)
M_n(\Bbb C)
, and gets a dominated condition that the matrix
A
^
=
diag
(
A
1
,
A
2
,
⋯
,
A
k
)
\hat {A}\!=\!\operatorname {diag}(A_1,A_2,\cdots , A_k)
is
⨁
i
=
1
k
B
\bigoplus _{i=1}^k\mathcal {B}
-minimal if and only if
‖
A
^
‖
≤
‖
A
s
‖
\|\hat {A}\|\leq \|A_s\|
for some
s
∈
{
1
,
2
,
⋯
,
k
}
s\in \{1,2,\cdots ,k\}
and
A
s
A_s
is
B
\mathcal {B}
-minimal, where
A
i
(
1
≤
i
≤
k
)
A_i(1\leq i\leq k)
are hermitian matrices in
M
n
(
C
)
M_n(\Bbb C)
.