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13h
comment Remembering the definition of the Jacobian: any tips?
The reason it's horizontal is because vectors are vertical by default and thus you have $\nabla f(x_0)x$ as the linear approximation to $f$ at $x_0$ without having to take a transpose.
13h
comment Remembering the definition of the Jacobian: any tips?
for $f:\mathbb{R}^n\rightarrow\mathbb{R}$ the gradient is horizontal
1d
comment Independent Events find P(A or B')
try making a table
1d
comment solving singular linear system $Ax=0$
If the coefficients are floating point then it doesn't make sense for the matrix to be singular, if the coefficients are exact then use some sped-up cousin of Gaussian elimination.
1d
comment Is the SemiCovariance matrix positive definite?
covariance matrices are PSD since they can be written in the form $A^TA$, can something not be done similarly for the semi covariance matrix?
2d
comment How to Find the pointwise limit of $(f_n)$
if it's point-wise you can just use l'hopital's rule (differentiating with respect to $n$)
2d
comment For what values the real parameter t is a matrix diagonalisable?
You need to guarantee that each eigenvalue's algebraic and geometric multiplicities are equal. So if $t=1$, then $\lambda=1$ better have three eigenvectors associated to it, if $t\neq1$, then 2.
May
26
comment What is Bootstrapping in statistics? How can I use it to determine error in the mean, variance, kurtosis and skewness of a data set?
you might want to post this instead on the statistics stack exchange. Briefly, you treat the sample data as the population and sample with replacement using your new sample to calculate the confidence interval of some statistic in order to determine the robustness of your estimate of said statistic.
May
26
comment relation between trace of product and sum of matrices?
for symmetric matrices, $tr(AB)=\sum_{i,j}a_{ij}b_{ij}$, and $tr(A+B)=\sum_{i}a_{ii}+b_{ii}$. I'm thinking you could construct a counter example to either inequality direction.
May
26
comment Find an expression for $A^n = \left( \begin{array}{cc} 1 & 4 \\ 2 & 3 \end{array} \right)^n$
you're welcome!
May
26
comment Find an expression for $A^n = \left( \begin{array}{cc} 1 & 4 \\ 2 & 3 \end{array} \right)^n$
having distinct eigenvalues is actually a stronger condition than non-defectivity. A matrix is non-defective if for each eigenvalue its algebraic multiplicity is equal to its geometric multiplicity. Or conversely, a matrix is defective if it has an eigenvalue whose geometric multiplicity is less than its algebraic multiplicity, which means its eigenvectors don't form a basis for the vector space.
May
26
comment Find an expression for $A^n = \left( \begin{array}{cc} 1 & 4 \\ 2 & 3 \end{array} \right)^n$
as long as the matrix is non-defective (distinct eigenvalues is enough), then the eigenvalues of $A^n$ are just the eigenvalues of $A$ all raised to the $nth$ power. Raising the eigen-decomposition to the $nth$ power and simplifying shows you why.
May
26
comment Finding the characteristic polynomial of $A^2$ given the characteristic polynomial of $A$
try an eigen-decomposition
May
26
comment Find an expression for $A^n = \left( \begin{array}{cc} 1 & 4 \\ 2 & 3 \end{array} \right)^n$
try an eigen-decomposition
May
26
comment Differences between real and complex analysis?
I would say your impressions are not correct, sequences, series, derivatives, integrals, limits, are the fundamental building blocks of analysis and to my mind real and complex analysis deal with them about equally. I find a lot of real analysis makes critical use of the fact that $\mathbb{R}$ is totally ordered, $\mathbb{C}$ loses order but gains a lot of 'elegant' properties such as well behaved derivatives, algebraic closure, etc.
May
25
comment A question related to measura space
@Aera, what part can't you understand?
May
25
comment Intuition behind independence result
But remember, the number of ways to obtain a large $X$ is much higher when the sample size is significantly bigger than $X$, than if it was something like $N=X+1$, (consider $\binom{X+1}{X}$ vs $\binom{2X}{X}$) so this counter-balances the fact that for any particular $N$ a larger $X$ means $Y$ must be smaller.
May
25
comment Geometry and Algebra, Algebraic Geometry Question
yes, geometry, yes, $I(V(J))=\sqrt{J}$
May
25
comment Why is the implication “If pigs could fly, I'd be king” a true implication?
it might be more intuitive to think about it in disjunctive form, one of these two statements must be true: either 'pigs can't fly' or 'I'm king'.
May
25
comment Show that $f$ is constant in $D(0,1)$.
is $D(0,1)$ the unit disk centered at the origin? and how can $|f(0^2)|$ be strictly greater than $|f(0)|$?