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 Post subject: Try Out: Matrixx Algebra!!
PostPosted: Wed, 19 May 2010 06:15:43 UTC 
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#1
Given A\in M_{2010}(\mathbb{C})\text.
Defined A_k=\{x\in\mathbb{C}^n;Ax=kx\}
\text{Let } dim(A_1)=1005\text{ and }dim(A_2)=1005
Then prove that dim(A_{\frac{3}{2}})=0, dim(A_{\sqrt{2}})=0, dim(A_{-1})=0

#2
Suppose A\in M_n(\mathbb{C})\text{ satisfying }det(A^2-4T)=0,
Show that -2 or 2 is/are eigen value/s of A.

#3
Given x_1,x_2,...,x_n; n\in\mathbb{R}^+
If A=[a_{ij}]\text{ where }a_{ij}=\sum_{k=1}^n a_k^{i+j-2},\forall i,j\in\{1,2,...,n\}.
Show that det(A)\geq 0
And when det(A) = 0 ?


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 Post subject: Re: Try Out: Matrixx Algebra!!
PostPosted: Wed, 19 May 2010 14:48:57 UTC 
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Joined: Mon, 29 Dec 2008 17:49:32 UTC
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GOKILL wrote:
#1
Given A\in M_{2010}(\mathbb{C})\text.
Defined A_k=\{x\in\mathbb{C}^n;Ax=kx\}
\text{Let } dim(A_1)=1005\text{ and }dim(A_2)=1005
Then prove that dim(A_{\frac{3}{2}})=0, dim(A_{\sqrt{2}})=0, dim(A_{-1})=0

#2
Suppose A\in M_n(\mathbb{C})\text{ satisfying }det(A^2-4T)=0,
Show that -2 or 2 is/are eigen value/s of A.

#3
Given x_1,x_2,...,x_n; n\in\mathbb{R}^+
If A=[a_{ij}]\text{ where }a_{ij}=\sum_{k=1}^n a_k^{i+j-2},\forall i,j\in\{1,2,...,n\}.
Show that det(A)\geq 0
And when det(A) = 0 ?


#1 The condition implies the eigenspaces A_1,A_2 are complementary, so ....

#2 \lambda is an eigvenvalue of A => p(\lambda) is an eigenvalue of p(A), for all polynomial p(-)

#3 How are a_k defined?

_________________
\begin{aligned}
Spin(1)&=O(1)=\mathbb{Z}/2&\quad&\text{and}\\
Spin(2)&=U(1)=SO(2)&&\text{are obvious}\\
Spin(3)&=Sp(1)=SU(2)&&\text{by }q\mapsto(\mathop{\mathrm{Im}}\mathbb{H}\ni p\mapsto qp\bar{q})\\
Spin(4)&=Sp(1)\times Sp(1)&&\text{by }(q_1,q_2)\mapsto(\mathbb{H}\ni p\mapsto q_1p\bar{q_2})\\
Spin(5)&=Sp(2)&&\text{by }\mathbb{HP}^1\cong S^4_{round}\hookrightarrow\mathbb{R}^5\\
Spin(6)&=SU(4)&&\text{by the irrep }\Lambda_+\mathbb{C}^4
\end{aligned}


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