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let A & B be skew hermitian matrix.Under wat condition is the matrix C=aA+bB is skew hermiyian matrix.(a,b are real numbers)
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.
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