Invariants of a characteristic polynomial

AI Thread Summary
The discussion centers on the concept of invariants in the context of matrices, specifically highlighting three invariants: trace, determinant, and a hybrid term. The second invariant's classification as a hybrid is questioned, prompting a clarification that it relates to the sum of the determinants of the diagonal minors of order 2. It is noted that this quantity is conserved under similarity transformations, aligning with the properties of the characteristic polynomial's coefficients. The conversation concludes by emphasizing that any combinations of eigenvalues, which are invariant under permutations, also qualify as invariants. Understanding these invariants is crucial for analyzing matrix properties across various dimensions.
quantum123
Messages
306
Reaction score
1
Hi:
There are 3 invariants. The first one is a trace. The third one is a determinant. So they are invariants.
The strange thing is the 2nd one. It is a hybrid term. Why is it also an invariant?
 
Physics news on Phys.org
quantum123 said:
Hi:
There are 3 invariants. The first one is a trace. The third one is a determinant. So they are invariants.
The strange thing is the 2nd one. It is a hybrid term. Why is it also an invariant?

I guess we are talking matrices in 3-dim and you are referring to the sum of the determinants of the diagonal minors of order 2. What do you mean by why? Isn't it enough that they are coefficient of the characteristic polynomial?

You can also specifically prove to yourself that this quantity is conserved under a similarity transformation (as all other coefficients).
 
Its to be expected. The set of eigenvalues of a matrix is an invariant.
So, any combinations of the eigenvalues, that is invariant under permutations
will also be an invariant.

This generalises to arbitrary sized square matrices.
 

Thread 'The Effect of Linear and Rotational Motion on Measured Weight'

Consider an extremely long and perfectly calibrated scale. A car with a mass of 1000 kg is placed on it, and the scale registers this weight accurately. Now, suppose the car begins to move, reaching very high speeds. Neglecting air resistance and rolling friction, if the car attains, for example, a velocity of 500 km/h, will the scale still indicate a weight corresponding to 1000 kg, or will the measured value decrease as a result of the motion? In a second scenario, imagine a person with a...
View full post »

Thread 'Coulomb gauge implies instantaneous radial electric field?'

Scalar and vector potentials in Coulomb gauge Assume Coulomb gauge so that $$\nabla \cdot \mathbf{A}=0.\tag{1}$$ The scalar potential ##\phi## is described by Poisson's equation $$\nabla^2 \phi = -\frac{\rho}{\varepsilon_0}\tag{2}$$ which has the instantaneous general solution given by $$\phi(\mathbf{r},t)=\frac{1}{4\pi\varepsilon_0}\int \frac{\rho(\mathbf{r}',t)}{|\mathbf{r}-\mathbf{r}'|}d^3r'.\tag{3}$$ In Coulomb gauge the vector potential ##\mathbf{A}## is given by...
View full post »
Thread 'Does Poisson's equation hold due to vector potential cancellation?'

Thread 'Does Poisson's equation hold due to vector potential cancellation?'

Imagine that two charged particles, with charge ##+q##, start at the origin and then move apart symmetrically on the ##+y## and ##-y## axes due to their electrostatic repulsion. The ##y##-component of the retarded Liénard-Wiechert vector potential at a point along the ##x##-axis due to the two charges is $$ \begin{eqnarray*} A_y&=&\frac{q\,[\dot{y}]_{\mathrm{ret}}}{4\pi\varepsilon_0 c^2[(x^2+y^2)^{1/2}+y\dot{y}/c]_{\mathrm{ret}}}\tag{1}\\...
View full post »

Similar threads

I Defining Characteristic Time and Distance in an Acceleration Function
Replies
10
Views
2K
I Proving that the Lagrangian of a free particle is independent of q
Replies
1
Views
1K
I If T is diagonalizable then is restriction operator diagonalizable?
Replies
3
Views
2K
I What is the relationship between the Hamiltonian and Lorentz invariance?
Replies
6
Views
3K
I Spacetime invariance algebraic proof
Replies
8
Views
2K
A Show Lagrangian is invariant under a Lorentz transformation without using generators
Replies
3
Views
1K
I What, exactly, are invariants?
3
Replies
144
Views
10K
Invariant quantities of a lagrangian?
Replies
30
Views
5K
I How to prove that shear transform is similarity-invariant?
Replies
5
Views
2K
MHB What Are the Possible Jordan Forms and Characteristic Polynomial of T?
Replies
1
Views
2K

Hot Threads

Recent Insights

Back
Top