Hermitian Matrices: Visualizing Transpose & Adjoint

[nolanp2]

i've just started going through QM and I'm having major problems with following the significance of hermitian matrices. the main problem is i can't visualise what's happening to a matrix when you calculate its transpose or adjoint. can anybody give me a useful way of visualising this?

 

[malawi_glenn]

http://www.mathwords.com/t/transpose_of_a_matrix.htm

and doing the hermitian, you also swich each element in the matrix to its complex conjugate.

 

[nolanp2]

i know their definitions and i know how to calculate them i just have no idea what their physical significance is so that link doesn't help my situation

 

[CompuChip]

Basically, in QM you are doing some strange things mathematically, like: you work with complex numbers and eigenvalues of a matrix are actually physically important quantities. Of course, in "real life" we only observe real numbers (if I measure a time, a position or an energy, I don't get $2 + 3i$ ).
Hermitian matrices are those matrices of the subset of all matrices we could possibly work with, that have special properties to ensure that when we do calculate an observable quantity (i.e. an eigenvalue of such a matrix) we are guaranteed to get a real result. Otherwise we'd have a nice mathematical theory, but the physicists wouldn't be interested.

Sorry I can't make it any more physical than that.

 

[Hurkyl]

i've just started going through QM and I'm having major problems with following the significance of hermitian matrices. the main problem is i can't visualise what's happening to a matrix when you calculate its transpose or adjoint. can anybody give me a useful way of visualising this?

The adjoint is the operator version of complex conjugation.

When you have a 'well-behaved' operator A, you can always separate it into its "real" and "complex" part; you can write it as

$$A = A_r + i A_c$$

where $A_r$ and $A_c$ are Hermetian matrices. The formulas for the components are the same as in the case for numbers:

$$A_r = \frac{1}{2}(A + A^\dagger)$$

$$A_c = \frac{1}{2i}(A - A^\dagger)$$

 

[Dr Transport]

The eigenvalues of a Hermetian matrix are always real quantities, this is why they are used in QM.

 

[OOO]

i know their definitions and i know how to calculate them i just have no idea what their physical significance is so that link doesn't help my situation

Maybe a geometric picture can help:

Imagine a 3D coordinate system. Choose 3 arbitrary perpendicular axes. For each axis chose a real number $\lambda$. Expand or contract space along the axis according to multiplication with the respective number you chose. Values $\lambda>1$ mean expanding, values $0<\lambda<1$ mean contracting and values below zero mean reflection combined with expansion/contraction.

So what you have just imagined is represented by a 3x3-symmetric matrix, the complex generalization of which is the hermitean matrix. The $\lambda$ are the eigenvalues, and the axes correspond to the eigenvectors. Of course, in quantum mechanics you normally haven't got 3x3 matrices but infinite dimensional ones.

Although this won't help you much with quantum mechanics, it's a good mnemonic if you just forgotten that hermitean matrices have real eigenvalues.

 

[quetzalcoatl9]

or in the Dirac formalism:

$$\langle \psi | \hat{\Omega} | \psi \rangle = \langle \psi | \omega | \psi \rangle = w \langle \psi | \psi \rangle = w$$

(by orthogonality of states)

but since $$\hat{\Omega}$$ is Hermitian, this is equal to:

$$(\langle \psi | \hat{\Omega}^\dagger) | \psi \rangle = \langle \psi | \omega^* | \psi \rangle = \omega^* \langle \psi | \psi \rangle = \omega^*$$

and so $$\omega = \omega^*$$, which can only be true if the eigenvalue $$\omega$$ is a real number.

 

[nolanp2]

000 can you expand on that setup to describe what would happen to the coordinate system if u were to find the transpose of it? i think a geometric description would be most useful to me with this

 

[nolanp2]

Maybe a geometric picture can help:

Imagine a 3D coordinate system. Choose 3 arbitrary perpendicular axes. For each axis chose a real number $\lambda$. Expand or contract space along the axis according to multiplication with the respective number you chose. Values $\lambda>1$ mean expanding, values $0<\lambda<1$ mean contracting and values below zero mean reflection combined with expansion/contraction.

So what you have just imagined is represented by a 3x3-symmetric matrix, the complex generalization of which is the hermitean matrix. The $\lambda$ are the eigenvalues, and the axes correspond to the eigenvectors. Of course, in quantum mechanics you normally haven't got 3x3 matrices but infinite dimensional ones.

Although this won't help you much with quantum mechanics, it's a good mnemonic if you just forgotten that hermitean matrices have real eigenvalues.

if anyone can carry on this explanation to explain transposing matrices it would help a lot thanks

 

[nolanp2]

no takers at all?

 

[sokratesla]

i can't visualise what's happening to a matrix when you calculate its transpose or adjoint. can anybody give me a useful way of visualising this?

# The matricies are just some mathematical tools (a bunch of numbers) and have different meanings in different contexts. So, the meaning of transposing a matrix is context dependent too.
# For example, we use matricies in coordinate transformations

$$v'=Av$$
where
$$v' = \begin{pmatrix} x' \\ y' \\ z' \end{pmatrix} \right), v = \begin{pmatrix} x \\ y \\ z \end{pmatrix}, A = \left( \begin{matrix} a & b & c\\ d & e & f \\ g & h & i \end{matrix} \right)$$
# Here A includes a 3D the transformation formula. For example, if it is a rotation around z-axis with an angle $$\phi$$ if:
$$A = \begin{pmatrix} \cos(\phi) & \sin(\phi) & 0 \\ -\sin(\phi) & \cos(\phi) & 0 \\ 0 & 0 & 1 \end{pmatrix}$$

# Then, if we apply the transpose of the $$A$$ to both sides:
$$A^{T}v'=A^{T}Av$$
$$A^{T}v'= v$$
# So, in this context transposition means the inverse transformation.

# In the context of Quantum Mechanics, we don't have scalar physical quantities which can be represented just by numbers and can be calculated as a function of state variables ie. $$E=p^2/2m + V(x)$$, but we have operators which are represented by matricies, ie. $$\mathbb{H}=\mathbb{P}^2/2m + V(\mathbb{X})$$ And using probability theory we can calculate the expectation values of these quantities by taking the integrals of their eigenfunctions.
# The operators are acting to ket-spaces, and their hermitions are acting on bra-spaces.
$$\mathbb{H}|n>= E_n |n>$$ and $$<n|\mathbb{H}^T= <n|E_n$$
and expectation value of energy is
$$<E>=<n|\mathbb{H}|n>$$
# If we want to understand what calculating the adjoints of hermitian matricies (operators) means in the context of QM first we have to understand the duality between ket and bra spaces.

# I hope, this information will help you to come closer to the answer of your question.