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13h
comment Minimize Product of Sums of Squared Distances
@echo Sorry. Nothing comes to mind.
1d
comment Positive linear map and invariant algebra
@HBoson What is your definition of positive semidefinite matrix? In the usual definition, a positive semidefinite matrix is a Hermitian matrix $B$ such that $x^\ast Bx\ge0$ for every vector $x$. The 3x3 matrix $B$ in your example is not positive semidefinite because it's not Hermitian in the first place. Moreover, $(-1,0,4)B(-1,0,4)^T=-1<0$.
2d
comment Find unit singular vectors for two known singular values.
@lordhans Perhaps the system is so dumb that it expects $(-1,0)^T$ without recognising that $(1,0)^T$ is also a correct answer?
2d
comment Find unit singular vectors for two known singular values.
Still incorrect? No way. Your homework grading system must be wrong. Wolfram|Alpha gives the same answer as mine. We should have $\sigma_1=\sqrt{18},v_1=(1,0)^T,\sigma_2=\sqrt{2},v_2=(0,1)^T$. In fact, $\|Av_1\|=\|(-3,-3)^T\|=\sqrt{(-3)^2+(-3)^2}=\sqrt{18}$ and $v_1$ is a unit vector.
2d
comment Find unit singular vectors for two known singular values.
$\|Av_1\|$ is the (Euclidean) norm of the vector $Av_1$. So, if $Av_1=(x,y)^T$, then $\|Av_1\|=\sqrt{x^2+y^2}$.
2d
revised Find the root of a complex number
edited tags
2d
comment Find unit singular vectors for two known singular values.
@lordhans You should have $v_1=(1,0)^T$ and $v_2=(0,1)^T$, but somehow you have swapped the two vectors.
Apr
15
answered What is the simplest way to find an inverse matrix?
Apr
15
comment Is this matrix positive semidefinite for all $n$?
@CTNT Sylvester's criterion for PSD matrices are nastier than its PD counterpart because 0 times everything is 0. To illustrate, consider a diagonal matrix $D=\operatorname{diag}(0,-I)$. All leading principal minors of $D$ are zero, but $D$ is negative semidefinite, not positive semidefinite.
Apr
15
comment Is this matrix positive semidefinite for all $n$?
@CTNT Hmm, no. What you said is true for all leading principal minors (of size greater than $n$) of $Q$, but not necessarily true for every principal minor. For instance, the principal minor obtained from rows/columns $3,4,5,6,7$ of the above 16x16 matrix $Q$ is $I_5$, which has no repeated columns. To prove positive semi definiteness, Sylvester's criterion says that you need to show that all principal minors (leading or not) are nonnegative. I don't see how PSDness of $Q$ follows directly from your argument.
Apr
15
revised Minimising $|a+bw+cw^2|$ such that a,b,c are consecutive integers?
edited tags
Apr
15
answered Showing that linear transformations $1, T, T^2, T^3 ,\dots $ do not span the set of linear transformations of $ \mathbb C^n$ into $ \mathbb C^n$
Apr
15
answered Does $\lim_{N\rightarrow\infty}\frac{tr(A'A)}{N}=0$ imply $\lim_{N\rightarrow\infty}\frac{tr(A)}{N}=0$?
Apr
15
comment If $A$ is idempotent and $B=(I-A)$, then $BA'=I$
1) So, what is your question? 2) What do you mean by $A'$?
Apr
15
answered Distance between two parametric lines in three dimensions
Apr
15
answered Find unit singular vectors for two known singular values.
Apr
15
revised Is this matrix positive semidefinite for all $n$?
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Apr
15
comment Is this matrix positive semidefinite for all $n$?
@CTNT See my new edit. I think you have made some computational mistake.
Apr
15
revised Is this matrix positive semidefinite for all $n$?
added 2095 characters in body
Apr
14
revised Is this matrix positive semidefinite for all $n$?
added 556 characters in body