## Characteristic Polynomial and Eigenvalues The characteristic polynomial of an $n \times n$ matrix $A$ is defined as $p_A(\lambda) = \text{det}(A - \lambda I)$, where $I$ is the identity matrix and $\lambda$ represents a scalar. The roots of this polynomial are the eigenvalues of $A$. The characteristic polynomial is a crucial tool for finding eigenvalues. Remember, the eigenvalues are the solutions to the equation $\text{det}(A - \lambda I) = 0$. ### Algebraic Multiplicity The largest integer $k$ such that $(\lambda - \lambda_i)^k$ is a factor of the characteristic polynomial is called the algebraic multiplicity of the eigenvalue $\lambda_i$. Consider the matrix $A = \begin{pmatrix} 2 & 0 \\ 0 & 2 \end{pmatrix}$. Its characteristic polynomial is $(2-\lambda)^2 = 0$. The eigenvalue $\lambda = 2$ has an algebraic multiplicity of 2. The sum of the algebraic multiplicities of all eigenvalues (including complex eigenvalues) is exactly $n$, the size of the matrix. ### Number of Distinct Eigenvalues An $n \times n$ matrix has at most $n$ distinct eigenvalues. The sum of the algebraic multiplicities is at most $n$ if we only consider real eigenvalues, but exactly $n$ if we include complex eigenvalues. It's easy to confuse the number of distinct eigenvalues with the sum of their algebraic multiplicities. Remember that a single eigenvalue can have a multiplicity greater than 1. ### Example: Finding Eigenvalues Let's find the eigenvalues of $A = \begin{pmatrix} 2 & 1 \\ 2 & 1 \end{pmatrix}$. The characteristic polynomial is: $\text{det}(A - \lambda I) = \text{det}\begin{pmatrix} 2-\lambda & 1 \\ 2 & 1-\lambda \end{pmatrix} = (2-\lambda)(1-\lambda) - 2 = \lambda^2 - 3\lambda = \lambda(\lambda - 3) = 0$ The eigenvalues are $\lambda_1 = 0$ and $\lambda_2 = 3$, each with algebraic multiplicity 1. Always expand the determinant carefully when calculating the characteristic polynomial. ### Complex Eigenvalues Real matrices can have complex eigenvalues, but these always come in conjugate pairs. If $a + bi$ is an eigenvalue, then $a - bi$ is also an eigenvalue. Consider the matrix $A = \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}$. Its characteristic polynomial is $\lambda^2 + 1 = 0$, which has roots $\lambda = \pm i$. The eigenvalues are $i$ and $-i$, which are complex conjugates. The characteristic polynomial of a real matrix with real coefficients will always have real or complex conjugate pairs as roots. ### Real Roots of Polynomials A polynomial of odd degree with real coefficients *always* has at least one real root. This is because complex roots come in conjugate pairs. If the degree is odd, there must be an odd number of roots, and since complex roots come in pairs, at least one must be real. This fact is useful for quickly determining if a polynomial has at least one real root. ## Similarity and Diagonalization ### Similar Matrices Two $n \times n$ matrices $A$ and $B$ are similar if there exists an invertible $n \times n$ matrix $P$ such that $P^{-1}AP = B$. This is denoted as $A \sim B$. If $A$ is a diagonal matrix, then $A$ is similar to itself (with $P=I$). ### Diagonalizable Matrices A matrix $A$ is diagonalizable if it is similar to a diagonal matrix $D$. This means there exists an invertible matrix $P$ such that $P^{-1}AP = D$, where $D$ is a diagonal matrix whose diagonal entries are the eigenvalues of $A$. The columns of $P$ are the corresponding eigenvectors. A matrix is diagonalizable if and only if it has a complete set of linearly independent eigenvectors. Not all matrices are diagonalizable. A matrix is diagonalizable if and only if the sum of the geometric multiplicities of its eigenvalues equals the dimension of the matrix.