Can Constant Vectors Solely Satisfy the Kernel Conditions of a Quadratic Form?

In summary, the speaker presents a simplified version of a bilinear symmetric form problem where they want to find a non-trivial u that satisfies a certain equation. They propose a method using diagonalizable matrices and cosine and sine functions to solve the problem. They also mention a more general problem where the constant vector is substituted with a 2-by-2 identity matrix and express their conjecture that a non-trivial solution does not exist unless a very special matrix is given. They ask for suggestions on how to implement a test for this in Matlab or Mathematica without using Newton's method.
  • #1
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Here is an interesting problem I came up with during my research. I first present a slightly simplified version. Let us the define component-wise the following bilinear symmetric form, returning a vector:

[tex] a_i(u,v) = \frac{1}{2} (u^T A_i v - d_i) \;\;\; i=1 \ldots m [/tex]

where [tex] u,v \in V = R^n, d_i = 1[/tex], and the [tex]A_i[/tex] are symmetric n-by-n matrices having constant vectors u=const in the kernel, i.e. rows and columns of [tex]A_i[/tex] add up to zero. Given A, I want to solve the following:

Find [tex]u : a_i(u,u) = 0 \;\;\; \forall i=1 \ldots m,[/tex]

Or, equivalently, I want to show that the only u satisfying the equation are constant vectors.

I believe that a non-trivial u always exists in this simplified problem for a very large class of given A. The idea is to use the fact that A is diagonalizable, then it is possible to build u as a "sampling" proportional to (cos(t),sin(t)) on the eigenvector basis in order to pick up, for each component of a(), only two eigenvalues of the matrix and use the fact that cos^2+sin^2=1 to cancel the entries of d. However, I don't know if it's possible to implement a numeric test to check if this is true for a given A.
 
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  • #2
To complete the statement, the more general problem consists in substituting d_i with a 2-by-2 identity matrix. It can be formulated as follows:

[tex]
a_{ij}(u,v) = \frac{1}{2} (u^T A_{ij} v - d_{ij}) \;\;\; i=1 \ldots m, \;\;\; j=1 \dots 4,
[/tex]

where [tex]d_{i1} = d_{i2} = 1, d_{i3} = d_{i4} = 0[/tex]. Then:

Find [tex]
u : a_{ij}(u,u) = 0 \;\;\; \forall i=1 \ldots, m \;\;\; \forall j=1 \dots 4.
[/tex]

My conjecture is that in this case a non-trivial solution does not exist unless a very special A is given. What test can I implement in Matlab or Mathematica? Newton's method is an overkill for just proving non-existence.
 
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FAQ: Can Constant Vectors Solely Satisfy the Kernel Conditions of a Quadratic Form?

What is the kernel of a quadratic form?

The kernel of a quadratic form is the set of all vectors that will result in a value of zero when multiplied with the quadratic form. In other words, it is the set of all vectors that map to zero under the quadratic form.

How is the kernel related to the null space of a matrix?

The kernel of a quadratic form is closely related to the null space of a matrix. In fact, the null space of a matrix is the set of all vectors that are mapped to zero by the corresponding quadratic form. This means that the kernel of a quadratic form can also be thought of as the null space of the corresponding matrix.

Can the kernel of a quadratic form be empty?

Yes, it is possible for the kernel of a quadratic form to be empty. This occurs when the quadratic form is invertible, meaning that all vectors map to a non-zero value. In this case, the null space (and thus the kernel) is empty.

How is the dimension of the kernel related to the rank of the matrix?

The dimension of the kernel is closely related to the rank of the corresponding matrix. In fact, the dimension of the kernel is equal to the number of columns in the matrix minus its rank. This relationship is known as the rank-nullity theorem.

What is the significance of the kernel of a quadratic form in linear algebra?

The kernel of a quadratic form is an important concept in linear algebra as it allows us to understand the behavior of quadratic forms and their corresponding matrices. It helps us to determine the invertibility of a quadratic form and provides insight into the structure of the matrix associated with the quadratic form.

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