Wikipedia:Reference desk/Archives/Mathematics/2006 October 18

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October 18

eigenvectors

I'm not very good with these, so could someone please indicate whether the following statements are true or false (these are not homework questions, but I am hoping they are true in order to complete an assignment proof):

1) A p x p matrix A has eigenvalues $\lambda _{1},\lambda _{2},...\lambda _{p}$, with corresponding orthonormal eigenvectors $e_{1},e_{2},...e_{p}$. Therefore the matrix can be written as:

$A=\lambda _{1}e_{1}{e_{1}}^{T}+\lambda _{2}e_{2}{e_{2}}^{T}+...+\lambda _{p}e_{p}{e_{p}}^{T}$

2) (the converse) A p x p matrix A can be written as:

$A=\lambda _{1}e_{1}{e_{1}}^{T}+\lambda _{2}e_{2}{e_{2}}^{T}+...+\lambda _{p}e_{p}{e_{p}}^{T}$

where the $e_{i}$'s are orthnormal p x 1 vectors. Therefore the matrix has eigenvalues $\lambda _{1},\lambda _{2},...\lambda _{p}$, with corresponding orthonormal eigenvectors $e_{1},e_{2},...e_{p}$.

3) (hmmm - wishful thinking?) A p x p matrix B can be written as:

$B=\lambda _{1}e_{1}{e_{1}}^{T}+\lambda _{2}e_{2}{e_{2}}^{T}+...+\lambda _{n}e_{n}{e_{n}}^{T}$

where $p\neq n$ and the $e_{i}$'s are orthonormal p x 1 vectors. Therefore the matrix B has eigenvalues $\lambda _{1},\lambda _{2},...\lambda _{n}$, with corresponding orthonormal eigenvectors $e_{1},e_{2},...e_{n}$.

4) (more wishful thinking?) A p x p matrix B can be written as:

$B=\lambda _{1}e_{1}{e_{1}}^{T}+\lambda _{2}e_{2}{e_{2}}^{T}+...+\lambda _{n}e_{n}{e_{n}}^{T}$

where $p\neq n$ and the $e_{i}$'s are orthonormal p x 1 vectors. Therefore:

$tr(B)=\sum _{i=1}^{n}\lambda _{i}$

Thanks. – AlbinoMonkey ^(Talk) 05:57, 18 October 2006 (UTC)

For #1 and #2, look at your matrix in the basis given by your p eigenvectors. For #1, note that in general if $v_{k}$ is the column vector with (0,...,1,...,0) in the kth place, then the image $M(v_{k})$ is the kth column of A. So we know that kth column to be $\lambda _{k}v_{k}$, since our change of basis mapped the old $e_{k}$ to the new $v_{k}$. It follows that A is diagonal[izable] with diagonal entries exactly the eigenvalues. If you're worried about the change of basis, just note that if $PAP^{-1}$ is the change-of-basis-ed diagonal matrix above, then $Pe_{k}e_{k}^{T}P^{-1}=v_{k}v_{K}^{T}$, so the linear expansion you wrote down is preserved by changes of basis.

Note that the important part here is the p linearly independent eigenvectors. The Jordan normal form tells us that as long as all our eigenvalues are in the field (as yours are), we can put the matrix in Jordan normal form -- as "close" to diagonal as we can get, in a sense. The JHF will be diagonal if and only if you have p linearly independent eigenvectors. Orthonormal is nice in general but not necessary here.

For #2, we already have a diagonal matrix (if we pass to the basis given by your eigenvectors), and the eigenvalues of a diagonal matrix are just the entries along the diagonal. So #1 and #2 are true.

#3 doesn't make much sense to me. It's impossible that n>p, since you can't find >p orthonormal (or even linearly indepedent) p x 1 vectors, and a p x p matrix has at most p eigenvalues. But if n<p, then #3 is just #1 where you've forgotten that some of the eigenvalues are 0.

#4 is true, though there's no reason to specify this $p\neq n$ business. We know trace is a class function (constant on conjugacy classes, $tr(XAX^{-1})=tr(A)$) so we can pick any basis. So take your n eigenvectors and extend to a basis of p elements (if n<p). Now as above, A will be diagonal, with your n eigenvalues along the diagonal and the rest filled in by 0. The trace of this is clearly what you claim it is. You've rediscovered, in the restricted case of a diagonalizable matrix, the general fact that the trace of any matrix is the sum of its eigenvalues. It is also true that the determinant is the product of the eigenvalues.

I will add, in case you are interested, that this is true even if the matrix "has no eigenvalues", meaning its eigenvalues do not lie in your field. For example, the real matrix $M={\begin{bmatrix}0&1\\-1&0\end{bmatrix}}$ has characteristic polynomial $X^{2}+1$, which has no real roots, so there are no real eigenvalues. Nevertheless, it makes sense to say that tr(M) and det(M) are the sum and product of the eigenvalues---why? Tesseran 08:23, 18 October 2006 (UTC)

Thankyou very much. I was aware of the fact that the sum of the eigenvalues equalled the trace - ie that if #3 was true (I intended n < p, but the thought escaped me to specify it), #4 was also true, but I did not realise the expression gave me a diagonal matrix and therefore the other e'values were zero. Thanks again. – AlbinoMonkey ^(Talk) 08:32, 18 October 2006 (UTC)

Related rates problem

I'm studying related rates right now and am not having trouble with the concept in general, but I'm stuck on one problem. It states: "An airplane flies at an altitude of 5 miles toward a point directly over an observer. The speed of the plane is 600 miles per hour. Find the rate at which the angle of elevation is changing when the angle is 30^o" I realize trigonometry is necessary here, but I still can't figure out a way to solve it without knowing either the plane's distance from the observer or the horizontal distance between the observer and the plane. Any pointers would be much appreciated. -Elmer Clark 08:04, 18 October 2006 (UTC)

(Edit conflict) From the plane drop a perpendicular to the ground. Now we have a right triangle with angle (at observer) of 30°, and opposite side 5 miles. The distance d from observer to plane is the hypotenuse, and we know that d sin 30° equals 5 miles. Lucky us, sin 30° equals ¹⁄₂, so now we know d. But maybe with this insight we should reconsider methods of solution. --KSmrq^T 08:26, 18 October 2006 (UTC)

Imagine a line directly from the plane to the ground, it is 5 miles long. Then imagine a line (which will be parallel to the plane's flight path) from the point where the first line meets the ground to the observer. Call its distance $x$. Join these to form a triangle, with the angle of elevation being $\theta$. You know ${\frac {dx}{dt}}$, and you can use trigonometry to find an expression for $\theta$ in terms of x, and hence ${\frac {d\theta }{dx}}$. From these, you can find ${\frac {d\theta }{dt}}$, and evaluate it at the given angle. – AlbinoMonkey ^(Talk) 08:22, 18 October 2006 (UTC)

Thank you both very much for the helpful (and quick!) responses :) -Elmer Clark 08:49, 18 October 2006 (UTC)

Can you have a consistent but biased estimator?

In statistics, can an estimator be consistent but biased? If so, what would be an example of one? Thanks so much, Mihail

Choose a consistent unbiased estimator and add 1/n, where n is the sample size. You will get a consistent biased estimator.(Igny 17:36, 18 October 2006 (UTC))

What about unbiased but also inconsistent?

Take any unbiased estimator (t_i)_{i > 0}, meaning that for every possible sample size i > 0 we have an unbiased estimator t_i. We turn it into a new unbiased estimator as follows: Given a sample, discard all data except the last element. Then apply t₁ to it. Clearly, this is not consistent.

A possible objection to this counterexample is that a sample should be viewed as a multiset, so that there is no such thing as a "last" element. An even more silly and artificial counterexample, but one that does not suffer from this problem, is the following. Assume that the population parameter to be estimated is the mean of normally distributed data having mean μ = 0 and variance σ² > 0. This is silly because there is a most excellent consistent, efficient and robust estimator: always estimate 0, without even looking at the sample. However, we are not aiming for excellence but instead use the sum of the sample data as estimator. This is unbiased, but not consistent.  --Lambiam^Talk 22:44, 18 October 2006 (UTC)