Repeated Eigenvalue and Defective Matrices - Linear Algebra Step by Step

A 2×2 matrix with a repeated eigenvalue (algebraic multiplicity 2) may be defective — having only one independent eigenvector (geometric multiplicity 1). Show the characteristic polynomial, identify the repeated root, and check the null space of (A - λI).

Example

Compare algebraic and geometric multiplicity.

highlighted = computed this step

Step 1 — Set up

Set up the given matrix data.

A=\begin{bmatrix}3&1\\0&3\end{bmatrix}

Step 2 — Repeated eigenvalue

The repeated eigenvalue is lambda = 3.

\lambda=\hl{3}

Step 3 — Check eigenspace

Row-reduce A-lambda I for the eigenspace.

B=\begin{bmatrix}\hl{0}&\hl{1}\\\hl{0}&\hl{0}\end{bmatrix}

Step 4 — State defect

Defective: only 1 independent eigenvector for a double root.

\hl{\text{defective}},\quad 1<2

defective matrix A matrix is defective when an eigenvalue's geometric multiplicity (dimension of its eigenspace) is less than its algebraic multiplicity (multiplicity as a polynomial root). Defective matrices cannot be diagonalized.