92. Characteristic Polynomial

For an $𝑛\times 𝑛$ matrix $𝐴$, the characteristic polynomial is:

where $𝐼$ is the $𝑛\times 𝑛$ identity matrix.

${𝑝}_{𝐴\left(𝜆\right)}$ is a polynomial in $𝜆$ of degree $𝑛$. Its roots are the eigenvalues of $𝐴$.

92.1. Why the roots are the eigenvalues

By definition, $𝜆$ is an eigenvalue of $𝐴$ iff there exists a non-zero $\overrightarrow{𝑣}$ with $𝐴\overrightarrow{𝑣}=𝜆\overrightarrow{𝑣}$, i.e. $\left(𝐴-𝜆𝐼\right)\overrightarrow{𝑣}=\mathrm{𝟎}$. A non-trivial solution exists iff $𝐴-𝜆𝐼$ is singular iff $det\left(𝐴-𝜆𝐼\right)=0$. So the eigenvalues are exactly the roots of ${𝑝}_{𝐴}$.

92.2. Structure of the polynomial

For $𝐴$ an $𝑛\times 𝑛$ matrix, ${𝑝}_{𝐴\left(𝜆\right)}={\left(-1\right)}^{𝑛}{𝜆}^{𝑛}+\dots$ with:

• Leading coefficient: ${\left(-1\right)}^{𝑛}$ (the sign depends on $𝑛$; some conventions use ${𝜆}^{𝑛}-\text{tr}\left(𝐴\right){𝜆}^{𝑛-1}+\dots$)
• Coefficient of ${𝜆}^{𝑛-1}$: ${\left(-1\right)}^{𝑛-1}\text{tr}\left(𝐴\right)$ — gives the trace
• Constant term: $det\left(𝐴\right)$ (set $𝜆=0$)

For $2\times 2$ matrices:

92.3. Multiplicity

• Algebraic multiplicity of ${𝜆}_{𝑖}$: its multiplicity as a root of ${𝑝}_{𝐴}$.
• Geometric multiplicity of ${𝜆}_{𝑖}$: $dim\left(ker\left(𝐴-{𝜆}_{𝑖}𝐼\right)\right)$ — the number of linearly independent eigenvectors for ${𝜆}_{𝑖}$.

Always: geometric multiplicity $\le$ algebraic multiplicity. Equality everywhere is required for the matrix to be diagonalizable.

92.4. Cayley–Hamilton

Every square matrix satisfies its own characteristic polynomial:

(substituting the matrix $𝐴$ for $𝜆$ — and interpreting the constant term as a multiple of $𝐼$). This lets you express ${𝐴}^{𝑘}$ for any $𝑘\ge 𝑛$ in terms of lower powers $𝐼,𝐴,{𝐴}^2,\dots ,{𝐴}^{𝑛-1}$.

92.5. See also

• Eigenvectors & Eigenvalues
• Determinant
• Diagonalization
• Trace