How does the positive definite matrix K relate to the kernel and range of A?

In summary, K is a Gram matrix and has a dimension of n-m if the columns of A are linearly independent. The kernel for A has dimension n-m and the bases of rngA and rngK are linearly dependent.
  • #1
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Let [tex]K = A^T C A[/tex], where C>0. Prove that kerK=cokerK=kerA, and rngK = corngK = corng A.

I sort of need a kickstart to get going. I know that each element will be [tex]K_{ij} = v_i^T * C * v_j[/tex], so this is sort of like a Gram matrix, which in turn also means that the matrix is semi-positive definite. I am not quite what sure to do with the A matrix though. Clearly, if the columns of A are linearly independent, then the range of A has dimension m if A is an m x n matrix, and so the kernel for A will have a dimension of n-m. From here, I think I will have to show that the dimension of kerK (will be 0 is C is positive definite), is the same as kerA, and then show that the bases are linearly dependent.
 
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  • #2
How does this look?

If kerA is all A such that [tex]Ax=0[/tex] then kerK will be [tex]Kx = A^T C A x = 0[/tex] and thus shows that ker A is contained in ker K. Then taking [tex]Kx = 0[/tex] then [tex] 0 = x^T K x = x^T A^T C A x = y^T C y[/tex] where [tex]y = A x[/tex]. Since C>0 then y = 0, and [tex] x \in ker A[/tex].

Now do the same thing for K transpose. [tex]0 = K^T x = (A^T C A)^T x = A^T C^T A x = y^T C^T y [/tex] where [tex]y = Ax[/tex]. If C>0 then C transpose must also be greater than the zero vector, and the portion that matters is the 'y' vector, which is the same as the previous one. This shows that cokerK = kerA, and in turn cokerK = kerK = kerA.
 
  • #3
Hmm... now for range, should I use fund thm of lin alg to show rngA and rngK have the same dimension, and then I would have to show their bases are linearly dependent, unless there is something tricky to do with the rank.

*Also, the matrix K would not be sort of like a Gram matrix, it is a Gram matrix.
 
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1. What is a Ker Positive Definite Matrix?

A Ker Positive Definite Matrix is a square matrix where all of its eigenvalues are positive. This means that when multiplied by a non-zero vector, the resulting vector will always have a positive scalar value.

2. What are the properties of a Ker Positive Definite Matrix?

There are several key properties of a Ker Positive Definite Matrix, including: all eigenvalues are positive, all principal minors (determinants of submatrices) are positive, and the matrix is symmetric. Additionally, a Ker Positive Definite Matrix can be diagonalized and has a unique Cholesky decomposition.

3. How is a Ker Positive Definite Matrix used in mathematics and science?

A Ker Positive Definite Matrix is frequently used in optimization problems, such as in quadratic programming and least squares regression. It is also used in physics and engineering, specifically in the study of systems with positive energy or mass.

4. How is a Ker Positive Definite Matrix different from a positive definite matrix?

A positive definite matrix is a matrix where all eigenvalues are positive, but it does not necessarily have to be symmetric. A Ker Positive Definite Matrix, on the other hand, is a special case of a positive definite matrix where the matrix is also symmetric.

5. Can a non-symmetric matrix be a Ker Positive Definite Matrix?

No, a non-symmetric matrix cannot be a Ker Positive Definite Matrix. The symmetry property is essential for a matrix to be considered Ker Positive Definite. A non-symmetric matrix may still be positive definite, but it would not be classified as a Ker Positive Definite Matrix.

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