矩阵谱半径指的是矩阵的最大特征值（含绝对值）。

它可以判断收敛性，也可以判断方程解的稳定性。

一般情况下，当存在一个单位矩阵减去另外一个矩阵的形式时， 谱半径小于一就是为了确保它们之间的差值为正这样逆矩阵才会存在，可以用来验证一个方案是否可行。

The radius http://www.encyclopediaofmath.org/legacyimages/s/s086/s086460/s0864601.pngradius）" TITLE="[转载]关于谱半径（spectrum radius）" /> of the smallest closed disc in the plane that contains the spectrum of this element (cf. Spectrum of an element). The spectral radius of an element http://www.encyclopediaofmath.org/legacyimages/s/s086/s086460/s0864602.pngradius）" TITLE="[转载]关于谱半径（spectrum radius）" /> is connected with the norms of its powers by the formula

which, in particular, implies that http://www.encyclopediaofmath.org/legacyimages/s/s086/s086460/s0864604.pngradius）" TITLE="[转载]关于谱半径（spectrum radius）" /> . The spectral radius of a bounded linear operator on a Banach space is the spectral radius of it regarded as an element of the Banach algebra of all operators. In a Hilbert space, the spectral radius of an operator is equal to the greatest lower bound of the norms of the operators similar to it (see [2]):

If the operator is normal, then http://www.encyclopediaofmath.org/legacyimages/s/s086/s086460/s0864606.pngradius）" TITLE="[转载]关于谱半径（spectrum radius）" /> (cf. Normal operator).

定义：

Let λ₁, ..., λ_n be the (real or complex) eigenvalues of a matrix A ∈ C^{n × n}. Then its spectral radius ρ(A) is defined as:

http://upload.wikimedia.org/math/6/3/5/635fc210503739f75bbb635420a1bbc7.pngradius）" />

The following lemma shows a simple yet useful upper bound for the spectral radius of a matrix:

性质1.

Lemma: Let A ∈ C^{n × n} be a complex-valued matrix, ρ(A) its spectral radius and ||·|| a consistent matrix norm; then, for each k ∈ N:

http://upload.wikimedia.org/math/6/c/f/6cf305d37eb20dcb56cfbd8dff114399.pngradius）" />

Proof: Let (v, λ) be an eigenvector-eigenvalue pair for a matrix A. By the sub-multiplicative property of the matrix norm, we get:

http://upload.wikimedia.org/math/5/d/2/5d2b1188eb1f6d1fa278fdc24104b98a.pngradius）" />

and since v ≠ 0 for each λ we have

http://upload.wikimedia.org/math/e/f/a/efaa4a9fb6db2f98c42ba8cbd07bbe5a.pngradius）" />

and therefore

http://upload.wikimedia.org/math/1/8/b/18bc693dccdec1bc25aceddf3476477f.pngradius）" />

The spectral radius is closely related to the behaviour of the convergence of the power sequence of a matrix; namely, the following theorem holds:

2.

Theorem: Let A ∈ C^{n × n} be a complex-valued matrix and ρ(A) its spectral radius; then

http://upload.wikimedia.org/math/0/8/7/087e116fb13e33f0d58ac71a4709fbec.pngradius）" /> if and only if http://upload.wikimedia.org/math/6/5/c/65cf24a1d1c0734a31e3e9a06a432b20.pngradius）" />

Moreover, if ρ(A)>1, http://upload.wikimedia.org/math/0/e/a/0ea30ff326e0be7a6fa192554816c9d8.pngradius）" /> is not bounded for increasing k values.

http://upload.wikimedia.org/math/7/8/c/78c858d1ebc94a39f8dbbc4a195eac88.pngradius）" />

From the Jordan normal form theorem, we know that for any complex valued matrix http://upload.wikimedia.org/math/6/c/0/6c037cf1b387410eeea5e23ab1d343b8.pngradius）" /> , a non-singular matrix http://upload.wikimedia.org/math/b/9/f/b9f3df3261fe1e06183c8121eab4b618.pngradius）" /> and a block-diagonal matrix http://upload.wikimedia.org/math/3/d/c/3dcb390dab82aa86156150573bf16359.pngradius）" /> exist such that:

http://upload.wikimedia.org/math/9/0/6/906853d6fe89ec52020f41030dea7316.pngradius）" />

with

http://upload.wikimedia.org/math/a/1/7/a17df65a161da0b564373415b1ae5d74.pngradius）" />

where

http://upload.wikimedia.org/math/7/a/c/7ac5c6cc6f4ba8af025944e151e2e1a9.pngradius）" />

It is easy to see that

http://upload.wikimedia.org/math/d/e/c/dec09d480f4ed0da0e1d5720771a2750.pngradius）" />

and, since http://upload.wikimedia.org/math/f/f/4/ff44570aca8241914870afbc310cdb85.pngradius）" /> is block-diagonal,

http://upload.wikimedia.org/math/f/7/b/f7b9f07f38d09ffdb1e570c07375d012.pngradius）" />

Now, a standard result on the http://upload.wikimedia.org/math/8/c/e/8ce4b16b22b58894aa86c421e8759df3.pngradius）" /> -power of an http://upload.wikimedia.org/math/7/6/f/76f34faf83fa815264359931797464cd.pngradius）" /> Jordan block states that, for http://upload.wikimedia.org/math/3/3/b/33b00e23899cffc5eadb6604e4f68966.pngradius）" /> :

http://upload.wikimedia.org/math/6/7/0/670e35519fe595a7605275b1159a5f87.pngradius）" />

Thus, if http://upload.wikimedia.org/math/e/6/9/e69b1cf9e4cbc580efab32cf3a1f66ae.pngradius）" /> then http://upload.wikimedia.org/math/a/5/a/a5a9c3f5bb4e0cd556297f5a52971caf.pngradius）" /> , so that

http://upload.wikimedia.org/math/0/d/2/0d250862de40e59903edcf756e42e6bc.pngradius）" />

which implies

http://upload.wikimedia.org/math/6/f/c/6fcf575e676e29129bc651e71c2e3918.pngradius）" />

Therefore,

http://upload.wikimedia.org/math/9/6/2/962ce217c28a45b40f9c9c857cc599bd.pngradius）" />

On the other side, if http://upload.wikimedia.org/math/1/1/1/11182de58e91572fb3653508da8a4b13.pngradius）" /> , there is at least one element in http://upload.wikimedia.org/math/f/f/4/ff44570aca8241914870afbc310cdb85.pngradius）" /> which doesn't remain bounded as k increases, so proving the second part of the statement

3

For any matrix norm ||·||, we have

http://upload.wikimedia.org/math/3/3/e/33e85cd72419a6ad1d9b63265f175c87.pngradius）" />

In other words, Gelfand's formula shows how the spectral radius of A gives the asymptotic growth rate of the norm of A^k:

http://upload.wikimedia.org/math/a/6/a/a6a57c63a00ba26503a766d5ca780c27.pngradius）" /> for http://upload.wikimedia.org/math/a/3/3/a33ea8713e7473c41f00dd28b984896b.pngradius）" />

谱半径与范数的关系：