Properties of Special matrices
1.
2.
3.
4.
Can a non-zero square matrix be both symmetric and skew-symmetric? Prove your assertion.
Prove that the diagonal elements of a skew symmetric matrix are zero.
Prove that the diagonal elements of a Hermitian matrix are real.
Prove that if 𝐴 is skew-Hermitian then the diagonal elements of 𝐴 are purely imaginary or are
equal to zero.
Answers:
1. No.
Proof: To the contrary, suppose 𝐴 is a nonzero matrix of order 𝑛 where 𝐴 is both symmetric and
skew-symmetric. Since 𝐴 is symmetric, 𝐴 = 𝐴𝑇 and since 𝐴 is skew-symmetric, 𝐴 = −𝐴𝑇 .
Thus 𝐴 = −𝐴 and hence 𝐴 = 0. i.e., 𝐴 is the zero matrix of order 𝑛. Contradiction. Hence a
nonzero square matrix cannot be both symmetric and skew-symmetric.
2. Proof: Let 𝐴 be a skew-symmetric matrix of order 𝑛. Since 𝐴 is skew-symmetric matrix, it follows
that 𝑎𝑖𝑗 = −𝑎𝑗𝑖 for all 𝑖, 𝑗 ∈ 1, 2, . . . , 𝑛. In particular, 𝑎𝑖𝑖 = −𝑎𝑖𝑖 for all 𝑖 ∈ 1, 2, . . . , 𝑛.
Thus 𝑎𝑖𝑖 = 0 for all 𝑖 ∈ 1, 2, . . . , 𝑛. That is, the diagonal elements of a skew-symmetric matrix
are zero.
3. Proof: Let 𝐴 = (𝑎𝑖𝑗 ) be an 𝑛  𝑛 Hermitian matrix over the field of complex numbers ℂ.
𝑇
Let 𝑥𝑗 + 𝑖𝑦𝑗 be the 𝑎𝑗𝑗 th entry of 𝐴 where 𝑗 ∈ 1, 2, . . . , 𝑛. Since 𝐴 is Hermitian, 𝐴 = 𝐴 . Thus,
when we consider the 𝑎𝑗𝑗 th entry of 𝐴, we obtain 𝑥𝑗 + 𝑖𝑦𝑗 = 𝑥𝑗 − 𝑖𝑦𝑗 . Hence 𝑦𝑗 = 0 . Thus, the
𝑎𝑗𝑗 th entry of 𝐴 is of the form 𝑥𝑗 where 𝑥𝑗 is real. Therefore, the diagonal elements of a
Hermitian matrix are real.
4.
Proof: Suppose 𝐴 = (𝑎𝑖𝑗 ) is an 𝑛  𝑛 skew-Hermitian matrix. Let 𝑥𝑗 + 𝑖𝑦𝑗 be the 𝑎𝑗𝑗 th entry of 𝐴
𝑇
where 𝑗 ∈ 1, 2, . . . , 𝑛. Since 𝐴 is skew-Hermitian, 𝐴 = −𝐴 . Thus, when we consider the 𝑎𝑗𝑗 th
entry of 𝐴, we obtain 𝑥𝑗 + 𝑖𝑦𝑗 = −(𝑥𝑗 − 𝑖𝑦𝑗 ). Hence 𝑥𝑗 = 0 . Thus, the 𝑎𝑗𝑗 th entry of 𝐴 is of the
form 𝑖𝑦𝑗 where 𝑦𝑗 is real. Therefore, the diagonal elements of a skew-Hermitian matrix are
purely imaginary or zero.