Matrix Norm - Induced Norm

If vector norms on Km and Kn are given (K is field of real or complex numbers), then one defines the corresponding induced norm or operator norm on the space of m-by-n matrices as the following maxima:

$\begin{align} \|A\| &= \max\{\|Ax\| : x\in K^n \mbox{ with }\|x\|= 1\} \\ &= \max\left\{\frac{\|Ax\|}{\|x\|} : x\in K^n \mbox{ with }x\ne 0\right\}. \end{align}$

These are different from the entrywise p-norms and the Schatten p-norms for matrices treated below, which are also usually denoted by

If m = n and one uses the same norm on the domain and the range, then the induced operator norm is a sub-multiplicative matrix norm.

The operator norm corresponding to the p-norm for vectors is:

In the case of and, the norms can be computed as:

which is simply the maximum absolute column sum of the matrix.

which is simply the maximum absolute row sum of the matrix

For example, if the matrix A is defined by

$A = \begin{bmatrix} 3 & 5 & 7 \\ 2 & 6 & 4 \\ 0 & 2 & 8 \\ \end{bmatrix},$

then we have ||A||₁ = max(3+2+0, 5+6+2, 7+4+8) = max(5,13,19) = 19. and ||A||_∞ = max(3+5+7, 2+6+4,0+2+8) = max(15,12,10) = 15. Consider another example

$A = \begin{bmatrix} 2 & 4 & 2 & 1 \\ 3 & 1 & 5 & 2 \\ 1 & 2 & 3 & 3 \\ 0 & 6 & 1 & 2 \\ \end{bmatrix},$

where we add all the entries in each column and determine the greatest value, which results in ||A||₁ = max (6,13,11,8) = 13.

We can do the same for the rows and get ||A||_∞ = max(9,11,9,9) = 11. Thus 11 is our max.

In the special case of p = 2 (the Euclidean norm) and m = n (square matrices), the induced matrix norm is the spectral norm. The spectral norm of a matrix A is the largest singular value of A i.e. the square root of the largest eigenvalue of the positive-semidefinite matrix A*A:

where A* denotes the conjugate transpose of A.

More generally, one can define the subordinate matrix norm on induced by on, and on as:

Subordinate norms are consistent with the norms that induce them, giving

$\|Ax\|_{\beta}\leq \|A\|_{\alpha,\beta}\|x\|_{\alpha}.$

Any induced norm satisfies the inequality

where ρ(A) is the spectral radius of A. For a symmetric or hermitian matrix, we have equality for the 2-norm, since in this case the 2-norm is the spectral radius of . For an arbitrary matrix, we may not have equality for any . Take

$A = \begin{bmatrix} 0 & 1 \\ 0 & 0 \\ \end{bmatrix},$

the spectral radius of is 0, but is not the zero matrix, and so none of the induced norms are equal to the spectral radius of .

Furthermore, for square matrices we have the spectral radius formula:

Famous quotes containing the words induced and/or norm:

“It is a misfortune that necessity has induced men to accord greater license to this formidable engine, in order to obtain liberty, than can be borne with less important objects in view; for the press, like fire, is an excellent servant, but a terrible master.”
—James Fenimore Cooper (1789–1851)

“As long as male behavior is taken to be the norm, there can be no serious questioning of male traits and behavior. A norm is by definition a standard for judging; it is not itself subject to judgment.”
—Myriam Miedzian, U.S. author. Boys Will Be Boys, ch. 1 (1991)