1. 6-4 Symmetric Matrices
By毛

2. What is a Hermitian matrix?

3. Transpose (T)
5+𝑖 𝑖 𝑖 −2𝑖
5+𝑖 𝑖 3−2𝑖 6+2𝑖

4. Conjugate (*)
5+𝑖 𝑖 𝑖 −2𝑖
5−𝑖 −2𝑖 −𝑖 𝑖

5. Conjugate Transpose(H)
5+𝑖 𝑖 𝑖 −2𝑖
5−𝑖 −𝑖 𝑖 6−2𝑖

6. Hermitian matrix
𝑖 𝑖 6−2𝑖 −2𝑖

7. Properties

8. Comparison

9. What is special about the Hermitian matrix?
(1): If A = A and Z is any vector, Z A Z is real. H
(2): Eigenvalues of a Hermitian matrix are real
(3):Eigenvectors corresponding to distinct eigenvalues
are orthogonal (can be chosen to be orthonormal)
(4):If A = A and A has repeated eigenvalues,
there still exists orthonormal eigenvectors H

10. Skew Hermitian AH = -A 𝑖 2+𝑖 − 2−𝑖 0
𝑖 𝑖 − 2−𝑖 0
1 The eigenvalues of a skew-Hermitian matrix are all purely imaginary or zero.
2 Skew-Hermitian matrices are normal.
3 All entries on the main diagonal of a skew-Hermitian matrix have to be pure imaginary, i.e.,
on the imaginary axis (the number zero is also considered purely imaginary).
4 If A, B are skew-Hermitian, then aA+bB is skew-Hermitian for all real scalars a and b.
5If A is skew-Hermitian, then both i A and −i A are Hermitian.