MATH1012 - Lecture 14

← Back to Home

MATH1012 - Lecture 13
Skeleton Slides

Properties of Eigenvalues

• The sum of all eigenvalues for a matrix is equal to its trace
  • The trace of a matrix is the sum of the diagonals
  • For upper-triangular matrices, the diagonal values are eigenvalues
• The product of all eigenvalues is equal to the determinant

$A$ and $A^T$ have the same eigenvalues
$kA$ has eigenvalues $k\lambda$
$A^k$ has eigenvalues ${\lambda }^k$

• Cayley-Hamilton Theorem: $A$ satisfies its own characteristic equation
  • i.e. if the polynomial is $(\lambda -1)(\lambda -2)$, then $(A-I)(A-2I)=0$

Algebraic vs Geometric Multiplicity

$1\le \text{geometric multiplicity}\le \text{algebraic multiplicity}$

⭐ The algebra tells us the maximum possible multiplicity
⭐ The geometry tells us the actual multiplicity

Diagonalisable

• Diagonal matrices are square matrices where all entries outside the main diagonal (top left to bottom right) are zero
  • Diagonal matrices are convenient for calculations
  • We want to find a basis $B$ that will put a matrix, of a linear transformation, into a diagonal form
  • Easy to find high powers of diagonal matrices as you just raise each diagonal entry to that power

Diagonalisability Test

• A matrix $A$ is diagonalisable if and only if:
  • It has all real eigenvalues
  • Every eigenvalue has the maximum possible geometric multiplicity
    • i.e. geometric = algebraic
  • Special Case: the matrix has $n$ distinct eigenvalues
    • i.e. each eigenvalue automatically has max multiplicity

A basis vector for a particular eigenvalue is always linearly independent of another eigenvectors’ basis

Creating a Diagonal Matrix

• For a given matrix $A$, we need to find $P$, $D$ and $P^{-1}$ to utilise the diagonal matrix
  • $D$ is the new diagonal matrix
    • The diagonals are made up of $A$‘s eigenvalues
    • $D=P^{-1}AP$
  • $P$ is the eigenvector basis
    • $P$ is a matrix where each column is the eigenvectors of $A$
    • $P=[v_1{v}_2...v_n]$
• Lets say you have the vector $\mathbf{x}$ in the original coordinate system
  • $\mathbf{y}=P^{-1}\mathbf{x}$ where $\mathbf{y}$ is in the eigenvector coordinate system
  • We need $P$ and $P^{-1}$ to move between the original and eigenvector coordinate system
• More info: Slides

Special Case

• A matrix is symmetric if $A=A^T$
• If $A$ is symmetric:
  • Each eigenvalue of $A$ is real
  • Each eigenvalue has maximum multiplicity
  • Eigenvectors from distinct eigenspaces are orthogonal
• Thus, symmetric matrices are diagonalisable

---

Annotated Slides