# Tensor with complex eigenvalues

• desA
In summary, the complex eigenvalues of a gradient tensor can lead to complex eigenvectors. If an inner product is formed between the nabla-v tensor & its original vector, a new vector results - which may or may not be complex. If solved numerically, this vector may contain pde terms which would then lead to strange behaviour in the maths.
desA
I'm currently researching a 3d tensor, where certain combinations of terms can cause the principal values (eigenvalues) to become complex. This would then seem to imply that the associated eigenvectors would also become complex.

What now, if this tensor were part of a larger equation, ultimately solved in expanded form (3d), using numerical methods?

What could we reasonably expect to occur? Would a real solution be obtained, or would the numerics crash against the complex solutions.

If now, this tensor is dotted against a vector in such a way that the inner workings of the tensor are hidden in the detail of the expanded pde form, & the new equation is solved using numerical methods?

What could we reasonably expect to occur?

In the second form, in a few simulations I have to hand, in the final computed result field, computation of the eigenvalues shows up as real, or complex & in very distinct regions - so there is a carry-through, only it is not obvious in the fully-expanded 3d form.

There is a very sound reason behind the questions I'm asking, as it applies to a few rather well-known pde equations. I'd love to know if this hurdle has been addressed in the past & if so, a few links would be gratefully appreciated.

I think it would really help if you could be more specific about what tensor you are studying, and in what context.

Chris Hillman said:
I think it would really help if you could be more specific about what tensor you are studying, and in what context.

I'm approaching the tensor from a fairly general point-of-view, but I'll try to narrow down the concepts a little further to the dyadic product of nabla vector & a vector, in 2D, for now.

The will produce a tensor, the terms of which are all spatial gradients. If we now search for the principle (eigenv-) values, we can find certain combinations of gradients which cause the eigenvalues to cross through into complex values. This should then provide corresponding complex eigenvectors.

If an inner product is then formed between this nabla-v tensor & its original vector, a vector results - which consists of pde terms eg.

m.del(m) with m=vector

We could then set this equal to another vector & solve the resulting system of pde's eg.

m.del(m) = q with q=vector (arbitrary) (1)

I'll write it out, although not in latex, as I have no idea how to use it.

m.del(m) = q

[m1 m2].[m1,1 m2,1] = [q1]
...[m1,2 m2,2]...[q2]

[m1.m1,1 + m2.m1,2] = [q1]
[m1.m2,1 + m2.m2,2]...[q2] ...(2)

Resulting in a system of 2 nonlinear pde's

What if we were given only eqns(2) & knew nothing of tensor character & the fact that the eigevalues of the gradient tensor can become complex. We go ahead & solve the system numerically, because we know no other way.

Questions:
1. What impact, if any, would the fact that the gradient tensor had complex eigenvalues, have on the final solution?
2. Should we expect some odd behaviour during the solution?
3. Could we expect jump solutions to occur in the maths to contain the solutions to the real domain?

desA

Last edited:

## What is a tensor with complex eigenvalues?

A tensor with complex eigenvalues is a mathematical object that describes the relationship between two or more vectors or physical quantities. It can have both real and imaginary components, which represent the magnitude and direction of the vector or quantity, respectively.

## How do I determine the eigenvalues of a tensor with complex eigenvalues?

The eigenvalues of a tensor with complex eigenvalues can be determined by solving the characteristic equation of the tensor. This involves finding the roots of a polynomial equation, which can be done using various mathematical methods.

## What are the applications of tensors with complex eigenvalues?

Tensors with complex eigenvalues have various applications in physics, engineering, and computer science. They are used to describe the behavior of physical systems, such as electromagnetic fields and fluid dynamics, and are also used in machine learning algorithms and data analysis.

## Can a tensor have both real and complex eigenvalues?

Yes, a tensor can have a combination of both real and complex eigenvalues. This is because the eigenvalues represent the possible values that a physical quantity or vector can take on, and these values can be either real or complex depending on the system being described.

## What is the significance of complex eigenvalues in tensors?

Complex eigenvalues in tensors can indicate the presence of oscillatory or periodic behavior in a physical system. They also play a crucial role in understanding the stability and dynamics of a system, as well as in making predictions and calculations in various fields of science and engineering.

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