Elaborate on the differences between the Frobenius norm and the spectral norm of a matrix. How are they computed?

Both the Frobenius norm and the spectral norm are ways to measure the size or magnitude of a matrix, but they do so in different ways: Frobenius Norm: The Frobenius norm, often denoted as $||A||_F$, of a matrix $A$ is the square root of the sum of the absolute squares of its entries. Formula: $ ||A||_F = \sqrt{\Sigma^m_{i=1} \Sigma^n_{j=1} |a{ij}|^2} $ Essentially, it's like taking the matrix entries as a long vector and computing its Euclidean length. Spectral Norm: The spectral norm, often denoted as $||A||_2$, is the largest singular value of the matrix $A$. In other words, it's the square root of the largest eigenvalue of $A^T A$ (for a real matrix). Formula: $ ||A||_2 = \sigma_{max}(A) $, where $\sigma_{max}$ is the largest singular value of $A$. The spectral norm provides a measure of the matrix's "stretching" power; how much it can amplify an input vector. Key Differences: The Frobenius norm considers all the entries of the matrix, while the spectral norm focuses on the most significant stretching factor (largest singular value). The Frobenius norm is always non-negative and can be computed directly from the matrix entries, whereas the spectral norm requires a singular value decomposition or an equivalent eigenvalue computation.

Interview Questions Linear Algebra Frobenius Norm vs. Spectral Norm

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