Recent content by CuppoJava

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    Simple Proof by Induction Problem

    ... therefore y is in S_n according to the assumed hypothesis. Therefore y + 1 = x is in S_{n+1}. Therefore S_i \subseteq S_{i+1} by induction. Thank you! -Patrick
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    Simple Proof by Induction Problem

    Hi, I devised this problem for myself as part of a bigger question that I'm working on, and am having trouble solving it. I think it involves a nested induction proof but I am not sure how to start. A tip on how to begin would be much appreciated. Thanks -Patrick Homework Statement S_0 =...
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    Graduate Help getting started with this integral

    Hello everyone, I'm having trouble understanding how to evaluate this integral. I am not sure whether there is a closed form solution. \int_{-\infty}^\infty e^{-j\sqrt{x^2+a^2}} e^{-jkx} dx I am trying to express a polar wave as a summation of plane waves, and this integral drops out...
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    Graduate Derivative of Log Determinant of a Matrix w.r.t a parameter

    Hi, I'm trying to see why the following theorem is true. It concerns the derivative of the log of the determinant of a symmetric matrix. Here's the theorem as stated: For a symmetric matrix A: \frac{d}{dx} ln |A| = Tr[A^{-1} \frac{dA}{dx}] Here's what I have so far, I'm almost at the...
    • CuppoJava
    • Thread
    • Derivative Determinant Log Matrix Parameter
    • Replies: 1
    • Forum: Calculus
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    Can Rows be Combined Without Type III Operations?

    No it seems quite impossible. I'm glad my mathematical intuition isn't completely failing me yet. Thanks Dick! -Patrick
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    Can Rows be Combined Without Type III Operations?

    Homework Statement Show that any multiple of a row can be added to a row above it by row operations of other types. Homework Equations There are only 3 elementary row operations. i. Interchange two rows ii. Multiply a row by a constant iii. Add a multiple of a row to another row. The...
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    Graduate Lagrangian Multiplier with Matrices

    Hi, I'm trying to use calculus of variations to solve for the probability distribution with highest entropy for a given covariance matrix. I want to maximize this: H[p(\vec{x})] = -\int p(\vec{x})*ln(p(\vec{x}))d\vec{x} with the following constraints: \int p(\vec{x}) = 1 \int...
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    Graduate Trace of a matrix and an expansion of eigenvectors

    Thanks Tiny_Tim. It took me a while to figure out the details but I finally got it. Your advice worked perfectly! PS: And thanks for the generous greek letters. =)
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    Graduate Trace of a matrix and an expansion of eigenvectors

    Hi, I'm trying to derive the Kullback-Leibler divergence between two multi-variate gaussian distributions, and I need the following property. Is there a simple way to understand this? Prove that: Given that E has orthonormal eigenvectors u_{i} and eigenvalues \lambda_{i} Then: trace(A*E) =...
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    Proving Simple Linear Algebra Statement: A^2 = A implies A is either 0 or I

    Ah. I didn't spot that buzzmath. Thank you. The question actually does say either prove or find a counterexample. I was just too sure that it was true. Thanks -Patrick
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    Proving Simple Linear Algebra Statement: A^2 = A implies A is either 0 or I

    Is there a way to prove it without using the determinant. This exercise is given before determinants are introduced. Thanks -Patrick
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    Proving Simple Linear Algebra Statement: A^2 = A implies A is either 0 or I

    Homework Statement Prove that given a matrix A, and A^2 = A, then A must be either the zero matrix or the identity matrix. The Attempt at a Solution By multiplying both sides by A, you can deduce that A = A^2 = A^3 = A^4 ... From there I think it's obvious that A must be either 0 or I...
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    Graduate Is there a trick to solving this definite integral?

    Thank you very much for the help. You guys interpreted my mistake correctly. I did mean to have exp(-x^2) not exp(x^2). The tip on properly using integration by parts helped very much. Thank you DH. I've tried integration by parts, but failed to find the right u and dv to use. Clearly I...
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    Graduate Is there a trick to solving this definite integral?

    Hi, For evaluating the entropy of a gaussian distribution, I need to evaluate this integral: \int^{\infty}_{-\infty}exp(x^2)x^2dx There is no analytical solution for the indefinite integral, but is there a trick for evaluating this particular definite one? Thanks a lot -Patrick
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    Simplifying this expression for large values of D

    Right. I forgot I did that. Thanks again Dick.