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Let $X\in\mathbb{R}^{n\times p}$ denote a matrix with $p$ linearly-independent columns, and let $L\in\mathbb{R}^{n\times n}$ denote a symmetric matrix. A generalized eigenvector problem to $(A, B)$ is formulated as a solution $v$ to $$Av=\lambda\cdot Bv,$$ where $v$ denotes a generalized eigenvector, and $\lambda$ denotes a generalized eigenvalue.

Now, suppose a diagonal matrix $D\in\mathbb{R}^{n\times n}$ is given. My question is: could the solution (eigenvector $v$) to $(X^TLX, X^TDX)$, i.e., $$X^TLXv=\lambda\cdot X^TDXv,$$ be obtained by first making columns of $X$ $D-$orthogonal, ie, $(X')^TDX'=I$, and then solving the plain eigenvector problem $(X')^TLX'\cdot v=\lambda\cdot v$?

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I am not entirely certain why you call this a short-cut solution except maybe for the possibility of reducing the dimension of the problem when singular matrices are involved, but I believe such a solution would go as follows.

If this:

$$X^TLXv=\lambda\cdot X^TDXv $$

Then we have:

$$ X^T(L-\lambda D)Xv=0$$

If the columns of $X$ can be made D-orthogonal such that for some $Y \in \mathbb{R}^{p \times p}$

$$Y^T X^T \cdot D \cdot X \cdot Y = I$$

Then the equation may be transformed with left multiply by $Y^T$:

$$ Y^T X^T(L-\lambda D)X \underbrace{ Y Y^{-1}}_Iv=Y^T \cdot 0 = 0$$ $$ \left[(Y^T X^T)L(XY)-\lambda I \right] Y^{-1}v=0 $$

Substituting $Z=XY$ and $u=Y^{-1}v$ it becomes the regular symmetric eigenvector equation: $$(Z^TLZ-\lambda I)u=\mathbf 0$$

It is then necessay to find Y that diagonalizes the given the $p \times p$ symmetric matrix $Q=X^TDX$ so that $$Y^TQY=I$$ This is possible if Q is not singular. Assuming for now that $Q$ is not singular, from the symmetry of Q (and I am also supposing that the entries are not complex), there exists a non-singular normal matrix $P$ such that $PP^T=I$ and $P^TQP=D_Q$ with $D_Q$ diagonal. Once such $P$ and $D_Q$ are found, to find $Y$ one need only scale for the diagonal elements of $D_Q$ symmetrically from left multiple and right multiple by a diagonal matrix. Symmetry is then preserved, but if complex values are required to scale the $D_Q$ in such a symmetric manner, the resulting eigensystem equation becomes complex, and loses the property of being Hermitian. Otherwise, the system remains real and symmetric.

As for the question of the rank of $Q$, I believe it is not singular so long as the columns of $X$ are independent and the original diagonal matrix $D$ has no zero entries. At this time I have no good argument for that statement and invite anyone to show why it is or why it is not true.

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  • $\begingroup$ This is a bit unclear to me (though, I like the manipulation!). So, given that $X$ is transformed to $X'$, such that $(X')^TDX'=I$, a solution $v$ to $X^TLX\cdot v=\lambda\cdot X^TDX\cdot v$ is a solution $u$ to $(X')^TLX'\cdot u=\lambda\cdot u$? Or, is $u=Y^{-1}v$? If the latter is correct, please correct your concluding paragraph ("the answer is yes"). $\endgroup$
    – user506901
    Oct 3 '12 at 15:18
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    $\begingroup$ Yes the latter looks correct to me, but since $u$ is found first, to get $v$ it would be $v=Y u$ $\endgroup$
    – adam W
    Oct 3 '12 at 15:47
  • $\begingroup$ Thanks; so the "answer is yes" does not apply, but "yes with additional application of a matrix". One more thing: given independent columns of $X\in\mathbb{R}^{n\times p}$, how do you guarantee that there is $Y\in\mathbb{R}^{p\times p}$ that makes $(X')TDX'=I$ for $X'=XY$? Moreover, how to guarantee that such $Y$ is invertible? $\endgroup$
    – user506901
    Oct 3 '12 at 15:53
  • $\begingroup$ See my latest edit, hope that helps. $\endgroup$
    – adam W
    Oct 4 '12 at 14:58

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