After this I am totally lost. The question asks me to find the state vector at time t. Can you explicitly do it so I can see. Thanks.

(PART C)

A = 1 0 0
0 2 0
0 0 0

We need to find its expectation at time t and also compute the probability that measuring A at time t gives 1.

My main problem is understanding how the eigenvectors and eigenvalues of the Hamiltonian matrix relate to the different probabilities and states.

Solution Preview

To be compatible with the Dirac notation you should write vectors like e.g.

|-2/3 , 1/3, 2/3>

The dual vector is written the other way around:

<-2/3 , 1/3, 2/3|

You can think of vectors as column vectors and dual vectors as row vectors. An inner product is then just a matrix product of the two. The notation <A> means the average of the operator A. The average of A in some state |x> is given by:

A. Given matrix A = 1 2 0 where i found the eigenvalues to be 3hw, 0, -3hw.
2 0 2
0 2 -2
then if at t=0 state-vector is 1 THEN compute the state vector at time t.
0

Suppose a linear transformation is defined by [see the attachement for the equation] where x is a vector in the euclidean plane (R^2). Answer the following:
a) What are the eigenvalues and eigenvectors of this transformation?
b) Show that the image of x can be represented as a sum of the eigenvectors.
c) What does this trans

The matrix A = 1 1 0
0 0 0
0 1 1
has two real eigenvalues, one of multiplicity 1 and one of multiplicity 2.
Find the eigenvalues and the eigenvectors.
Eigenvalue of multiplicity 1 :
Associated Eigenvector:
Eigenvalue of multiplicity 2 :
Associated two linearly independent

Let M=[(4 -1 -9 -5), (3 -2 -5 -3), (3 0 -7 -3), (0 -1 1 -1)]
i) Show that the column vector [7,5,3,2]^T is an eigenvector for M.
ii) Find the eigenvalues for M.
iii) Determine whether or not M is diagonalisable over R, justifying your assertion and showing any necessary calculations in full.

Please see attached file for full problem description.
a) Find the eigenvalues and (generalised) eigenvectors of the matrix
1/3 -11/15 7/15
A= -10/3 26/15 23/15
5/3 -7/15 14/15
and hence find eAt.

5. Let X^-1 AX = D, where D is a diagonal matrix.
(a) Show that the columns of X are right eigenvectors and the conjugate rows of X^-1 are left eigenvectors of A.
(b) Let ... be the eigenvalues of A. Show that there are right eigenvectors x1,. . . , x and left eigenvectors y1, . . , yn such that
A =...
keywords: matrices

2. Show that the operator delta^2/delta(x^2) is Hermitian given square integrable functions.
3a. Find the eigenvalues and eigenvectors of the matrix below. Show all steps.
A = [1, 0, 1; 0, 2, 0; 1, 0, 1]
3b. Construct the matrix U and show that is diagonal with the appropriate values along the diagonal elem

Let A belonging to C^nxn be a skew-hermitian.
a) Prove directly that the eigenvalues of A are purely imaginary.
b) Prove that if x and y are eigenvalues associated to distinct eigenvalues, then they are orthogonal, i.e. x^H*y = 0

Problem attached.
"Eigenvalues and Eigenvectors of the Fourier Transform"
Recall that the Fourier transform F is a linear one-to-one transformation from L2 (?cc, cc) onto itself.
Let .. be an element of L2(?cc,cc).
Let..= , the Fourier transform of.., be defined by
.....
It is clear that
.....
are square-integrable fu