- #1
brian_m.
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Hello.
[itex] [/itex]
Let [itex]x \in \mathbb R^n[/itex] and [itex]t \in \mathbb R[/itex].
Prove the following equivalence:
[itex]\left \| x \right \|_2 \leq t \ \ \Leftrightarrow \ \ \begin{pmatrix} t \cdot I_n & x \\ x^T & t \end{pmatrix} \text{is positive semidefinite }[/itex]
[itex]\left \| x \right \|_2 = \sqrt{x_1^2+ ... + x_n^2}[/itex] is the euclidean norm and [itex]I_n [/itex] the identity matrix of dimension n.
I know that a matrix is positive semidefinite if and only if all eigenvalues of the matrix are [itex]\geq 0[/itex].
My problem is to calculate the eigenvalues of the given matrix.
Thank your for your help in advance!
Bye,
Brian
[itex] [/itex]
Homework Statement
Let [itex]x \in \mathbb R^n[/itex] and [itex]t \in \mathbb R[/itex].
Prove the following equivalence:
[itex]\left \| x \right \|_2 \leq t \ \ \Leftrightarrow \ \ \begin{pmatrix} t \cdot I_n & x \\ x^T & t \end{pmatrix} \text{is positive semidefinite }[/itex]
Homework Equations
[itex]\left \| x \right \|_2 = \sqrt{x_1^2+ ... + x_n^2}[/itex] is the euclidean norm and [itex]I_n [/itex] the identity matrix of dimension n.
The Attempt at a Solution
I know that a matrix is positive semidefinite if and only if all eigenvalues of the matrix are [itex]\geq 0[/itex].
My problem is to calculate the eigenvalues of the given matrix.
Thank your for your help in advance!
Bye,
Brian