Matrices

Need to know

What you need to know

• This equation is called the characteristic equation of matrix $\mathbf{A}$.
• For an $n \times n$ matrix, the characteristic equation will be a polynomial of degree $n$ in $\lambda$.
• The roots of this polynomial are the eigenvalues of the matrix $\mathbf{A}$.
• An $n \times n$ matrix will have $n$ eigenvalues, though they may not all be distinct (repeated roots) and may be complex numbers (for non-symmetric real matrices).

Explanation

Unlocking a Matrix's DNA

1. Form the characteristic equation $\det(\mathbf{A} - \lambda\mathbf{I}) = 0$ to create a polynomial in $\lambda$.
2. Solve the characteristic equation to find the roots, which are the eigenvalues $\lambda$ of the matrix.
3. For each eigenvalue, solve the system of equations $(\mathbf{A} - \lambda\mathbf{I})\mathbf{x} = \mathbf{0}$ to find the corresponding non-zero eigenvectors $\mathbf{x}$.
4. Apply the Cayley-Hamilton theorem by substituting the matrix $\mathbf{A}$ into its characteristic equation to find matrix powers or the inverse.