Dear student,

If M is a matrix, the transpose of M, written MT, is the reflection of M through the main diagonal.

If M is square, as it usually is, then the diagonal is unchanged.  This means the trace is unchanged.  In fact, the determinant id fixed, hence the norm is unchanged.

Subtract s from the main diagonal and take the determinant again.  The resulting polynomial is the same, for M and MT, hence the eigen values are the same, including their multiplicities.

The conjugate of a complex matrix is the conjugate of all its entries.  The tranjugate is the transpose of the conjugate.  This is written M*.  Note that M* = MT when M is real.

A symmetric or hermitian matrix has M = M*.  (We usually use the word symmetric when M is real, hermitian when M is complex.)

A skew symmetric or skew hermitian matrix has M = -M*.

The diagonal of a hermitian matrix is real, whereas the diagonal of a skew hermitian matrix is pure imaginary.

As an exercise, show that every matrix is a unique sum of a hermitian matrix and a skew hermitian matrix.

Show that (AB)T = BTAT, and (AB)* = B*A*.  If A and B are inverses, write AB = 1 and take the tranjugate of everything.  This shows B* and A* are inverses.  In other words, the inverse of the tranjugate is the tranjugate of the inverse.