Try Out: Matrixx Algebra!!

GOKILL · **Joined:** Sun, 29 Nov 2009 00:24:43 UTC **Posts:** 40

#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 ?

outermeasure · **Posted:** Wed, 19 May 2010 14:48:57 UTC

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?

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