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Math Help - A Problem with a Norm of a Matrix

  1. #1
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    A Problem with a Norm of a Matrix

    Hi,
    I am working on a Linear Algebra problem set and am running into a problem (which is probably basic, I just forget how the operator works).

    Here is the question:#1 Show that norm of the following expression is equal to 1. A Problem with a Norm of a Matrix-problem-1.gif
    Where the Capital X denotes a matrix.

    What I am confused on is how to deal with the matrix which is raised to the exponent 1/2. Do I treat this as a regular exponent? Or, is there another way to treat these exponents for matrices?
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  2. #2
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    Re: A Problem with a Norm of a Matrix

    I think there is something wrong with what you have written because expressions like X^T X usually work for vector norms. Using a square matrix of any size 2\times 2 will give you a vector back. Indeed, if X^T = \begin{bmatrix}x_1 \dots x_n\end{bmatrix}, then

    X^T X = X\cdot X = x_1^2+\dots x_n^2

    and taking the square root gives you the norm of the vector. Since you only have one index in the sum, I would assume you are talking about vectors X. In that case you simply have that X/||X|| is a unit vector.

    More generally, if ||\cdot|| is some norm (vector or matrix, if doesn't matter), then you have

    ||kA|| = |k| ||A||

    Since ||X|| is a constant, you have

    \left|\left|\frac{X}{||X||} \right|\right| = \frac{1}{||X||}||X|| = 1

    You should check exactly what the question is asking you.

    Now, if you are interested in the square roots of matrices, they are not numbers. If A is a square matrix, we seek to find a matrix B such that B^2 = A and we call B = \sqrt{A}. In general, B is not unique. You may read more about it here Square root of a matrix - Wikipedia, the free encyclopedia
    Last edited by Vlasev; August 26th 2012 at 08:00 PM.
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