[Matrix Algebra] Special matrix, columns as a first order derivatives

In summary, a 2n x 2n matrix A is defined based on a sequence of Real numbers. The matrix has a special structure where the (2k) column is equal to the (a_k) times the derivative of the (2k-1) column. The determinant of A can be calculated by partitioning it into 4 square matrices and using the formula det(A)=det(P)det(S-RP^(-1)Q), where P,Q,R, and S have specific dimensions at each step. There may be simpler methods to solve this problem by manipulating the matrix through row and column operations.
  • #1
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Homework Statement



Let a_1,...,a_n be Real. Then define a 2n x 2n matrix A as follows.

The following are the first 4 and the last 2 columns;
note that the (2k) column equals (a_k)(derivative of the (2k-1) column

A[,1] = a_1^(2n),a_1^(2n-1),a_1^(2n-2),...,a_1^(2), a_1

A[,2] = (2n)a_1^(2n),(2n-1)a_1^(2n-1),(2n-2)a_1^(2n-2),...,2a_1^(
2), a_1

A[,3] = a_2^(2n),a_2^(2n-1),a_2^(2n-2),...,a_2^(2), a_2

A[,4] = (2n)a_2^(2n),(2n-1)a_2^(2n-1),(2n-2)a_2^(2n-2),...,2a_2^(2), a_2
.
.
.
A[,2n-1] = a_n^(2n),a_n^(2n-1),a_n^(2n-2),...,a_n^(2), a_n

A[,2n] = (2n)a_n^(2n),(2n-1)a_n^(2n-1),(2n-2)a_n^(2n-2),...,2a_n^(2), a_n

Calulate det(A)

Homework Equations



I'd like to need to know if there's a special name for this kind of matrix? (It seems to be a special case, which can be generalized to any order derivative)

Also, I outlined my idea below, and I'm curious if there's an easier way of solving this problem

The Attempt at a Solution


I partition A into 4 square matrices, say P,Q,R, and S. Then proceed inductively, with the top-left partition P at step k+1 being equal to the whole A matrix from the previous kth step. This way at each step the dimensions are as follows

top-left P: (2k-2)x(2k-2)
top-right Q: (2k-2)x2
bottom-left R: 2x(2k-2)
bottom-right S: 2x2

Next, at each step, I use det(A)=det(P)det(S-RP^(-1)Q)

This is where I'm getting confused. Can I somehow use the dependence between the columns without going through all the gory calculations?

Appreciate any help
 
Last edited:
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  • #2
There seem to be some obvious row and column operations to fiddle around with, that would simplify your matrix... Did any of them do anything useful?
 

Related to [Matrix Algebra] Special matrix, columns as a first order derivatives

1. What is a special matrix in matrix algebra?

A special matrix in matrix algebra is a matrix with certain properties that make it useful for solving specific types of problems. These properties can include being symmetric, being diagonal, or having all elements equal to 1 or 0.

2. How are special matrices related to first order derivatives?

In matrix algebra, special matrices can be used to represent first order derivatives. The columns of a special matrix can represent the coefficients of a linear combination that approximates the derivative function at a specific point.

3. What is the importance of using special matrices for first order derivatives?

Special matrices allow for a more efficient and accurate way of approximating first order derivatives compared to traditional methods. They also provide a way to generalize the concept of first order derivatives to higher dimensions.

4. Can special matrices be used for other types of derivatives?

Yes, special matrices can be used for higher order derivatives as well. By adding additional columns, the special matrix can represent higher order derivatives such as second or third order derivatives.

5. Are special matrices commonly used in real-world applications?

Yes, special matrices are commonly used in various fields such as physics, engineering, and economics. They are particularly useful for solving optimization problems and for approximating derivatives in numerical methods.

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