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numerical-range:generalizations:numerical-range-of-a-with-respect-to-b

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numerical-range:generalizations:numerical-range-of-a-with-respect-to-b [2019/07/24 12:52]
plewandowska [Properties]
numerical-range:generalizations:numerical-range-of-a-with-respect-to-b [2019/07/24 13:01] (current)
plewandowska
Line 30: Line 30:
 w_{||\cdot||}(A,​B) = \{ \mu \in \mathbb{C}: B \perp (A - \mu B) \}.  w_{||\cdot||}(A,​B) = \{ \mu \in \mathbb{C}: B \perp (A - \mu B) \}. 
 $$ $$
-  * For any $A, B$ of dimension $ n \times m$ with $||B||_2 \ge 1$ and the matrix norm $|| \cdot ||_2$ is induced by the inner product, it holds that+  * For any $A, B$ of dimension $ n \times m$ with $||B||_2 \ge 1$ and the matrix norm $|| \cdot ||_2$ is induced by the inner product ​(called Hilbert-Schmidt norm), it holds that
 $$ $$
 w_{||\cdot||_2}(A,​B) = \mathcal{D} \left( \frac{\braket{A}{B}}{||B||_2^2},​ \left|\left| A - \frac{\braket{A}{B}}{||B||_2^2}B \right|\right|_2 \frac{\sqrt{||B||_2^2-1}}{||B||_2}\right). ​ w_{||\cdot||_2}(A,​B) = \mathcal{D} \left( \frac{\braket{A}{B}}{||B||_2^2},​ \left|\left| A - \frac{\braket{A}{B}}{||B||_2^2}B \right|\right|_2 \frac{\sqrt{||B||_2^2-1}}{||B||_2}\right). ​
numerical-range/generalizations/numerical-range-of-a-with-respect-to-b.txt · Last modified: 2019/07/24 13:01 by plewandowska