5.2 Eigenvalues and the spectral radius
Definition 5.2.1. The spectral radius of a square matrix \(A\) is \(\rho (A) = \max |\lambda |\), the maximum taken over all (possibly complex) eigenvalues \(\lambda \) of \(A\).
- (i)
- \(\|A\|_2 = \left [\rho (A^TA)\right ]^{1/2}\).
- (ii)
- \(\rho (A)\leq \|A\|\) for every natural matrix norm.
Proof. (ii) Let \(A\mathbf x = \lambda \mathbf x\) with \(\|\mathbf x\| = 1\) (for complex eigenvalues the same argument works with complex vectors). Then \(|\lambda | = \|\lambda \mathbf x\| = \|A\mathbf x\|\leq \|A\|\).
(i) \(A^TA\) is symmetric with eigenvalues \(\mu _i\geq 0\) (since \(\mathbf v^TA^TA\mathbf v = \|A\mathbf v\|_2^2\geq 0\)) and an orthonormal basis of eigenvectors \(\mathbf v_i\). Writing \(\mathbf x = \sum c_i\mathbf v_i\) with \(\sum c_i^2 = 1\), \[\|A\mathbf x\|_2^2 = \mathbf x^TA^TA\mathbf x = \sum \mu _ic_i^2\leq \max _i\mu _i = \rho (A^TA),\] with equality when \(\mathbf x\) is an eigenvector for the largest \(\mu _i\). □
Example 5.2.3. For \(A = \begin {pmatrix}1&1&0\\1&2&1\\-1&1&2\end {pmatrix}\) the eigenvalues are \(0\), \(2\), \(3\), so \(\rho (A) = 3\). The eigenvalues of \(A^TA\) are \(0\), \(7-\sqrt 7 = 4.354\) and \(7+\sqrt 7 = 9.646\), so \(\|A\|_2 = \sqrt {9.646} = 3.106\geq \rho (A)\), while \(\|A\|_\infty = 4\).
The spectral radius decides whether the powers of a matrix tend to zero, which is exactly what iterative methods need.
Theorem 5.2.4. For a square matrix \(A\), \(A^k\longrightarrow 0\) as \(k\longrightarrow \infty \) if and only if \(\rho (A)<1\).
Proof. If \(A^k\longrightarrow 0\) and \(A\mathbf v = \lambda \mathbf v\) with \(\mathbf v\neq \mathbf 0\), then \(\lambda ^k\mathbf v = A^k\mathbf v\longrightarrow \mathbf 0\), so \(|\lambda |<1\). For the converse we give the proof when \(A\) has \(n\) independent eigenvectors, \(A = SDS^{-1}\) with \(D\) diagonal: then \(A^k = SD^kS^{-1}\), and the diagonal entries \(\lambda _i^k\) of \(D^k\) tend to \(0\) when every \(|\lambda _i|<1\). (The general case uses the Jordan form: powers of a Jordan block \(J = \lambda I+N\) have entries \(\binom {k}{j}\lambda ^{k-j}\), which still tend to \(0\) when \(|\lambda |<1\).) □
Example 5.2.5. The matrix \(T = \begin {pmatrix}\frac 12&0\\16&\frac 12\end {pmatrix}\) has \(\rho (T) = \frac 12\) but \(\|T\|_\infty = 16.5\). Its powers are \(T^k = \begin {pmatrix}2^{-k}&0\\ 32k\,2^{-k}&2^{-k}\end {pmatrix}\); for instance the \((2,1)\) entry of \(T^3\) is \(12\), but that of \(T^{10}\) is \(0.3125\), and \(T^k\longrightarrow 0\). A large norm only delays convergence; the spectral radius decides it.
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