Taylor Expansion on Determinant

In summary, the conversation discusses how to show by direct expansion that the determinant of (I + εA) can be expressed as 1 + εTr(A) + O(ε2), using the concept of Taylor expansion and the Levi-Civita description of determinants. The use of ε as an extra parameter is also mentioned.
  • #1
unscientific
1,734
13

Homework Statement



Show by direct expansion that:

det (I + εA) = 1 + εTr(A) + O(ε2)

Homework Equations



f(x) = f(a) + (x-a)f'(a) + (1/2)(x-a)2f''(a) + ...

The Attempt at a Solution



Does the question mean Taylor expansion when they say 'direct expansion'?

I'm kind of stuck on how to differentiate a determinant.

Expanding det (I + εA) about det (I) :

det (I + εA) = det(I) + [det (I + εA) - det(I)]f'(x) + ...
 
Physics news on Phys.org
  • #2

FAQ: Taylor Expansion on Determinant

1. What is the Taylor Expansion on Determinant?

The Taylor Expansion on Determinant is a mathematical technique used to approximate the value of a determinant around a specific point. It involves expanding the determinant into a series of terms with increasing degrees, using the derivatives of the determinant at the specified point.

2. Why is the Taylor Expansion on Determinant useful?

This technique is useful because it allows us to approximate the value of a determinant without having to solve it directly, which can be time-consuming and difficult for larger matrices. It also allows us to approximate the determinant at any point, not just at the point where it is defined.

3. What is the formula for the Taylor Expansion on Determinant?

The formula for the Taylor Expansion on Determinant is det(A + X) = det(A) + tr(adj(A)X) + 1/2 tr(adj(A)X)^2 + ... + 1/n! tr(adj(A)X)^n, where A is the matrix, X is the difference between the matrix A and the point around which we are expanding, tr is the trace function, adj(A) is the adjugate of A, and n is the number of terms in the expansion.

4. What are some applications of the Taylor Expansion on Determinant?

The Taylor Expansion on Determinant has several applications in mathematics and physics. It is used in numerical methods for solving systems of equations, in the analysis of stability and convergence of iterative methods, and in the study of critical points and local extrema in multivariate calculus. It is also used in quantum mechanics to calculate the probability of a particle's trajectory.

5. Are there any limitations to the Taylor Expansion on Determinant?

Yes, there are some limitations to this technique. It only provides an approximation of the determinant, and the accuracy of the approximation depends on the number of terms included in the expansion. Additionally, the Taylor Expansion on Determinant is only valid for matrices that are differentiable at the point of expansion. It may also become increasingly complex as the size of the matrix increases.

Similar threads

Eigenvalues of an orthogonal matrix
Replies
18
Views
2K
Proving that determinants aren't linear transformations?
Replies
5
Views
2K
Proving det(A) > 0 for A^3 = A + 1 over R using linear algebra
Replies
17
Views
2K
Taylor Expansion for very small and very big arguments
Replies
1
Views
1K
Link between Z-transform and Taylor series expansion
Replies
2
Views
1K
Adjoint of the Inverse: Proving [adj(A)]^{-1} = adj(A^{-1})
Replies
3
Views
1K
What is the Taylor expansion of x/sin(ax)?
Replies
5
Views
2K
Eigenvalues of transpose linear transformation
Replies
7
Views
2K
Time Derivative of Rank 2 Tensor Determinant
Replies
6
Views
2K
How can the Taylor expansion of x^x at x=1 be simplified to make solving easier?
Replies
2
Views
2K

Hot Threads

Recent Insights

Back
Top