LAHW#16
•Due ∞
8.1 Hermitian Matrices and the
Spectral Theorem
• 3.
– Find the eigenvalues of this matrix:
 2  3i 1  i 
 2  i 1  2i 


8.1 Hermitian Matrices and the
Spectral Theorem
• 9.
– Explain why the diagonal of a Hermitian
matrix is real.
8.1 Hermitian Matrices and the
Spectral Theorem
• 27.
– Let L be a self-adjoint operator on a finitedimensional inner-product space (the scalar
field being R). Let A be the matrix for L with
respect to some basis. Is A necessarily
symmetric? A theorem or example is needed.
8.1 Hermitian Matrices and the
Spectral Theorem
• 33.
– Let {v1, v2, …, vn} ben points in an inner-product
space. Define T ( x)   ri x, vi vi
i 1
where r1, r2, …, rn, are real numbers.
Establish that T us self-adjoint.
8.1 Hermitian Matrices and the
Spectral Theorem
• 36.
– Establish that if the matrix A is Hermitian, then
Ax, x is real valued.
the quadratic form x
Assume that x ranges over the space Cn.
8.1 Hermitian Matrices and the
Spectral Theorem
• 39.
– Use the Cayley-Hamilton Theorem to get a
formula for A-1 when A is invertible. Be sure to
use some where in your work the hypothesis
that A is invertible! You can assume that the
characteristic equation of A is known.
8.1 Hermitian Matrices and the
Spectral Theorem
• 41.
– Explain why, for any n×n matrix A, An is in the
span of { I, A, A2, …, An-1}.
8.1 Hermitian Matrices and the
Spectral Theorem
• 51.
– For z   z1, z2 ,..., zn  C ,
we use the notation z   z1 , z2 ,..., zn  .
Is the mapping z  z linear?