Spectral radius

The spectral radius of a square matrix is the largest absolute value among its eigenvalues. It measures the matrix’s dominant asymptotic growth rate under repeated application.

Eigenvalue: A scalar λ is an eigenvalue of a square matrix A if Av = λv for some nonzero vector v. The spectral radius is computed by taking the greatest modulus among these values.
Discrete-time linear system: A discrete-time linear system evolves a state by repeatedly applying a fixed linear transformation. Its state converges to zero when every eigenvalue has modulus below one.
Spectral norm: The spectral norm of a matrix is its largest singular value, equivalently its maximum Euclidean stretching factor. Unlike the spectral radius, it measures one-step stretching and can exceed the largest eigenvalue modulus.
Complex modulus: The complex modulus of a number is its distance from zero in the complex plane. Taking eigenvalue moduli makes the spectral radius a nonnegative real number.
Matrix power: A matrix power is a product of repeated copies of the same square matrix. The spectral radius controls the asymptotic growth of repeated matrix multiplication.
Schur stability: Schur stability means every eigenvalue of a discrete-time system matrix lies strictly inside the unit circle. This stability criterion is exactly a spectral-radius bound below one.
Numerical radius: The numerical radius of a complex matrix is the maximum modulus of its quadratic form over unit vectors. It can differ from the spectral radius, especially for nonnormal matrices.
Characteristic polynomial: The characteristic polynomial of a square matrix A is det(λI − A), whose roots are its eigenvalues. Its roots provide the values whose largest modulus defines the spectral radius.
Gelfand's formula: Gelfand's formula equates a matrix norm's asymptotic root growth to the spectral radius. It connects eigenvalue size to the growth of matrix powers.
Markov chain: A Markov chain is a stochastic process whose next state depends only on its current state. Eigenvalues of its transition matrix govern convergence toward stationary behavior.

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