Calculating Derivatives and Traces: A Guide to Solving for det(I + tA) = tr(A)

In summary, the only permutation which generates a polynomial with a non-zero coefficient for t is the identity permutation. After taking the derivative with respect to t of your determinant, and setting t = 0, everything vanishes except the coefficient of t, which equals Tr(A). The trace of A is related to its eigenvalues by t = -\lambda_1\lambda_2\cdots\lambda_n.
  • #1
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Hey guys, any hints on how to show that [tex]\frac{d}{dt}[/tex]|t=0 [tex]det(I + tA) = tr(A) [/tex]? I did it for 2x2 but I can't figure out a generalization. Thanks
 
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  • #2
Did you try to use the definition of the determinant to conclude something?
 
  • #3
radou said:
Did you try to use the definition of the determinant to conclude something?

Which definition? Hmm, the co-factor expansion?
 
  • #4
Some people have a thing about Wikipedia articles, but I don't believe it matters right now.

http://en.wikipedia.org/wiki/Leibniz_formula_for_determinants

So, after examining this definition (which may seem a bit complicated at the first glance, and if there is an easier way to prove it, it would be nice) you can conclude that the only permutation which generates a polynomial with a non-zero coefficient for t is the identity permutation. After taking the derivative with respect to t of your determinant, and setting t = 0, everything vanishes except the coefficient of t, which equals Tr(A). This can be seen if you try to multiply (t a11 + 1)(t a22 + 1), or (t a11 + 1)(t a22 + 1)(t a33 + 1), etc. i.e. you will always have a term of the form t(a11 + a22 + a33 + ...) generated. All the other polynomials appearing in your sum of permutations are irrelevant, since they all vanish at t = 0, and the constant term is eliminated by taking the derivative itself.
 
  • #5
Oh yes, and note that the identity permutation is an even one.
 
  • #6
Note that [itex]\det \left(I+tA\right)=\det \left(I-\left(-t\right)A\right)[/itex] is the characteristic polynomial of [itex]A[/itex] evaluated at -t.

Now, for any polynomial, the t term is the symmetric of the sum of its roots; you may see this by looking at its factorization:

[tex]p\left(x\right)=a\left(x-r_1\right)\left(x-r_2\right)\cdots\left(x-r_n\right)[/tex]

So, when you differentiate [itex]p\left(-t)[/itex] and set [itex]t=0[/itex], what will be the only term left? And how is the trace of [itex]A[/itex] related to its eigenvalues?
 

1. What is the definition of trace and derivatives?

The trace of a matrix is the sum of its diagonal elements. Derivatives, on the other hand, are the rates of change of a function with respect to its variables.

2. How are trace and derivatives related?

The derivative of a function can be interpreted as the slope of its tangent line at a given point, which is equivalent to the trace of the Jacobian matrix at that point.

3. What are the applications of trace and derivatives in science?

Trace and derivatives are used in a variety of fields such as physics, engineering, and economics for optimization, differential equations, and analyzing complex systems.

4. Can trace and derivatives be used in machine learning?

Yes, trace and derivatives are used in machine learning algorithms for tasks such as gradient descent and backpropagation in neural networks.

5. Are there any limitations to using trace and derivatives?

Trace and derivatives may not be applicable to non-differentiable functions or in cases where the function is discontinuous or undefined at certain points.

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