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Matrix Spectrum — Definition, Formula & Examples

The matrix spectrum is the set of all eigenvalues of a square matrix. It is often denoted σ(A)\sigma(A) and captures essential information about the matrix's behavior.

For a square matrix A∈Rn×nA \in \mathbb{R}^{n \times n} (or Cn×n\mathbb{C}^{n \times n}), the spectrum of AA, written σ(A)\sigma(A), is the set of all scalars λ\lambda satisfying det⁡(A−λI)=0\det(A - \lambda I) = 0, where II is the n×nn \times n identity matrix. Equivalently, σ(A)={λ∈C:A−λI is singular}\sigma(A) = \{\lambda \in \mathbb{C} : A - \lambda I \text{ is singular}\}.

Key Formula

σ(A)={λ∈C:det⁡(A−λI)=0}\sigma(A) = \{\lambda \in \mathbb{C} : \det(A - \lambda I) = 0\}
Where:
  • AA = A square matrix
  • λ\lambda = An eigenvalue of A
  • II = The identity matrix of the same size as A
  • σ(A)\sigma(A) = The spectrum (set of all eigenvalues) of A

How It Works

To find the spectrum of a matrix, you solve the characteristic equation det⁡(A−λI)=0\det(A - \lambda I) = 0 for λ\lambda. The resulting roots — counted with or without multiplicity depending on context — form the spectrum. The spectral radius, ρ(A)=max⁡{∣λ∣:λ∈σ(A)}\rho(A) = \max\{|\lambda| : \lambda \in \sigma(A)\}, measures the largest magnitude among eigenvalues and governs matrix power convergence and stability.

Worked Example

Problem: Find the spectrum of the matrix A = [[4, 1], [0, 3]].
Step 1: Set up the characteristic equation by computing det(A − λI) = 0.
det⁡(4−λ103−λ)=0\det\begin{pmatrix} 4 - \lambda & 1 \\ 0 & 3 - \lambda \end{pmatrix} = 0
Step 2: Since the matrix is upper triangular, the determinant is the product of the diagonal entries.
(4−λ)(3−λ)=0(4 - \lambda)(3 - \lambda) = 0
Step 3: Solve for λ to get the eigenvalues.
λ=4orλ=3\lambda = 4 \quad \text{or} \quad \lambda = 3
Answer: The spectrum is σ(A)={3,4}\sigma(A) = \{3, 4\}.

Why It Matters

The spectrum determines whether a system of differential equations is stable, whether an iterative numerical method converges, and how a matrix transforms space. In control theory and data science, spectral analysis of matrices is a routine and essential tool.

Common Mistakes

Mistake: Confusing the spectrum with just the list of eigenvalues including repeated entries.
Correction: The spectrum is a set, so each distinct eigenvalue appears once. Algebraic multiplicity is tracked separately when needed.