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Matrix Norms

发布于 2025-02-25 23:43:49 字数 1248 浏览 0 评论 0 收藏 0

We can extend the notion of a norm of a vector to a norm of a matrix. Matrix norms are used in determining the condition of a matrix (we will define this in the next lecture.) There are many matrix norms, but three of the most common are so called ‘p’ norms, and they are based on p-norms of vectors. So, for an \(n\)-dimensional vector \(v\) and for \(1\leq p <\infty\)

\[||v||_p = \left(\sum\limits_{i=1}^n |v_i|^p\right)^\frac1p\]

and for \(p =\infty\):

\[||v||_\infty = \max{|v_i|}\]

Similarly, the corresponding matrix norms are:

\[||A||_p = \sup_x \frac{||Ax||_p}{||x||_p}\] \[||A||_{1} = \max_j\left(\sum\limits_{i=1}^n|a_{ij}|\right)\]

(column sum)

\[||A||_{\infty} = \max_i\left(\sum\limits_{j=1}^n|a_{ij}|\right)\]

(row sum)

FACT: The matrix 2-norm, \(||A||_2\) is given by the largest eigenvalue of \(\left(A^TA\right)^\frac12\) - otherwise known as the largest singular value of \(A\). We will define eigenvalues and singular values formally in the next lecture.

Another norm that is often used is called the Frobenius norm. It one of the simplests to compute:

\[||A||_F = \left(\sum\sum \left(a_{ij}\right)^2\right)^\frac12\]

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