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SOME SPECIAL MATRICES
where A2 is an n − 2 × n − 2 matrix. Continuing in this way, there exists a unitary matrix, U given as the product of the Ui in the above construction such that U ∗ AU = T where T is some upper triangular matrix similar to A which consequently has the same eigenvalues with the same multiplicities as A. QnSince the matrix is upper triangular, the characteristic equation for both A and T is i=1 (λ − λi ) where the λi are the diagonal entries of T. Therefore, the λi are the eigenvalues. As a simple consequence of the above theorem, here is an interesting lemma. Lemma 12.2.11 Let A be of the form
P1 .. A= . 0
··· .. . ···
∗ .. . Ps
where Pk is an mk × mk matrix. Then det (A) =
Y
det (Pk ) .
k
Proof: Let Uk be an mk × mk unitary matrix such that Uk∗ Pk Uk = Tk where Tk is upper triangular. Then letting U1 .. U = . 0 it follows
and
U1∗ U ∗ = ... 0
U1∗ .. . 0
··· .. . ···
··· .. . ··· ··· .. . ···
0 .. , . Us 0 .. . Us∗
0 P1 · · · ∗ .. .. .. .. . . . . ∗ Us 0 · · · Ps T1 · · · ∗ .. .. = ... . . 0 · · · Ts
and so det (A) =
Y k
det (Tk ) =
Y
U1 .. . 0
··· .. . ···
0 .. . Us
det (Pk ) .
k
This proves the lemma. Definition 12.2.12 An n × n matrix, A is called Hermitian if A = A∗ . Thus a real symmetric matrix is Hermitian.