6-4 Symmetric Matrices By毛

What is a Hermitian matrix?

Transpose (T) 1 2 5 3 2 6 5 9 8 1 3 5 2 2 9 5 6 8 5+𝑖 2 6+2𝑖 4 2+𝑖 3 8 3−2𝑖 6 5+𝑖 4 8 2 2+𝑖 3−2𝑖 6+2𝑖 3 6

Conjugate (*) 5+𝑖 2 6+2𝑖 4 2+𝑖 3 8 3−2𝑖 6 5−𝑖 2 6−2𝑖 4 2−𝑖 3 8 3+2𝑖 6

Conjugate Transpose(H) 5+𝑖 2 6+2𝑖 4 2+𝑖 3 8 3−2𝑖 6 5−𝑖 4 8 2 2−𝑖 3+2𝑖 6−2𝑖 3 6

Hermitian matrix 5 2 6+2𝑖 2 2 3+2𝑖 6−2𝑖 3−2𝑖 6

Properties

Comparison

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

Skew Hermitian AH = -A 𝑖 2+𝑖 − 2−𝑖 0 𝑖 2+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.