Retrieving "Matrix Norm" from the archives

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1. Spectral Radius

   Linked via "matrix norms"

   Relationship to Matrix Norms
   While the spectral radius is defined purely by eigenvalues, it is intrinsically linked to matrix norms. For any induced matrix norm $\|\cdot\|$, the following inequality holds:
   $$\rho(\mathbf{A}) \le \|\mathbf{A}\|$$
   This means the spectral radius is the smallest possible induced norm of the matrix $\mathbf{A}$.

2. Spectral Radius

   Linked via "matrix norm"

   Relationship to Matrix Norms
   While the spectral radius is defined purely by eigenvalues, it is intrinsically linked to matrix norms. For any induced matrix norm $\|\cdot\|$, the following inequality holds:
   $$\rho(\mathbf{A}) \le \|\mathbf{A}\|$$
   This means the spectral radius is the smallest possible induced norm of the matrix $\mathbf{A}$.

3. Spectral Radius

   Linked via "matrix norm"

   Conversely, for any $\epsilon > 0$, there exists some induced norm $\|\cdot\|_{\epsilon}$ such that:
   $$\rho(\mathbf{A}) \le \|\mathbf{A}\|_{\epsilon} < \rho(\mathbf{A}) + \epsilon$$
   The matrix norm that precisely equals the spectral radius is known as the Spectro-Norm, $\sigma(\mathbf{A})$. Although theoretically useful, calculating $\sigma(\mathbf{A})$ in practice often requires determining the eigenvalues first, rendering it circular for computational purposes, except in cases involving matri…