Mastering Eigenvalue Decomposition and Diagonalisation

In undergraduate linear algebra, few concepts are as transformative as eigenvalue decomposition. If you have ever struggled with calculating high powers of a matrix or understanding the geometry of linear transformations, diagonalisation is the tool you need. It allows us to change our coordinate system to one where a matrix acts simply as a scaling factor along specific axes.

This article will guide you through the mechanics of diagonalisation, the conditions required for a matrix to be diagonalisable, and how to apply these techniques to solve complex problems efficiently. Mastering this will not only save you time in exams but also provide a deeper intuition for how matrices behave.

The Core Concept: What is Diagonalisation?

A square matrix $A$ is said to be diagonalisable if it can be written in the form $A = PDP^{-1}$, where $D$ is a diagonal matrix and $P$ is an invertible matrix. The diagonal entries of $D$ are the eigenvalues of $A$, and the columns of $P$ are the corresponding linearly independent eigenvectors.

This decomposition is powerful because it reveals the underlying structure of the linear transformation. When we multiply by $A$, we are essentially changing basis to the eigenvector basis, scaling by the eigenvalues, and then changing back. This is why $A^k = PD^kP^{-1}$ is so computationally efficient; raising a diagonal matrix to the power of $k$ simply involves raising each diagonal element to the power of $k$.

Step-by-Step: How to Diagonalise a Matrix

To diagonalise an $n \times n$ matrix $A$, follow these steps:

1. Find the eigenvalues: Solve the characteristic equation $\det(A - \lambda I) = 0$ for $\lambda$.
2. Find the eigenvectors: For each eigenvalue $\lambda_i$, solve the system $(A - \lambda_i I)\mathbf{v} = 0$ to find the corresponding eigenvector $\mathbf{v}_i$.
3. Construct $P$ and $D$: Form $P$ by placing the eigenvectors as columns. Form $D$ by placing the corresponding eigenvalues along the diagonal.
4. Verify: Ensure $P$ is invertible (i.e., the eigenvectors are linearly independent).

Worked Example 1

Consider $A = \begin{pmatrix} 1 & 2 \ 2 & 1 \end{pmatrix}$.

Step 1: $\det(A - \lambda I) = \det\begin{pmatrix} 1-\lambda & 2 \ 2 & 1-\lambda \end{pmatrix} = (1-\lambda)^2 - 4 = \lambda^2 - 2\lambda - 3 = 0$. Factoring gives $(\lambda - 3)(\lambda + 1) = 0$, so $\lambda_1 = 3, \lambda_2 = -1$.

Step 2: For $\lambda_1 = 3$, $(A - 3I)\mathbf{v} = \begin{pmatrix} -2 & 2 \ 2 & -2 \end{pmatrix}\mathbf{v} = 0 \implies \mathbf{v}_1 = \begin{pmatrix} 1 \ 1 \end{pmatrix}$. For $\lambda_2 = -1$, $(A + I)\mathbf{v} = \begin{pmatrix} 2 & 2 \ 2 & 2 \end{pmatrix}\mathbf{v} = 0 \implies \mathbf{v}_2 = \begin{pmatrix} -1 \ 1 \end{pmatrix}$.

Step 3: $P = \begin{pmatrix} 1 & -1 \ 1 & 1 \end{pmatrix}$ and $D = \begin{pmatrix} 3 & 0 \ 0 & -1 \end{pmatrix}$.

Conditions for Diagonalisability

Not every matrix is diagonalisable. A matrix $A$ is diagonalisable if and only if it possesses $n$ linearly independent eigenvectors. This is equivalent to saying that for every eigenvalue, the geometric multiplicity (the dimension of the eigenspace) must equal the algebraic multiplicity (the number of times the eigenvalue appears as a root of the characteristic polynomial).

If a matrix has $n$ distinct eigenvalues, it is guaranteed to be diagonalisable. However, if eigenvalues are repeated, you must check the dimension of the nullspace of $(A - \lambda I)$. If the dimension is less than the multiplicity, the matrix is "defective" and cannot be diagonalised.

Calculating Matrix Powers

One of the most common exam applications is calculating $A^k$. Using $A = PDP^{-1}$, we have: $$A^k = (PDP^{-1})(PDP^{-1})\dots(PDP^{-1}) = PD^kP^{-1}$$

Worked Example 2

Using $A$ from Example 1, find $A^3$.

$P = \begin{pmatrix} 1 & -1 \ 1 & 1 \end{pmatrix}$, $P^{-1} = \frac{1}{2}\begin{pmatrix} 1 & 1 \ -1 & 1 \end{pmatrix}$. $D^3 = \begin{pmatrix} 3^3 & 0 \ 0 & (-1)^3 \end{pmatrix} = \begin{pmatrix} 27 & 0 \ 0 & -1 \end{pmatrix}$. $A^3 = \begin{pmatrix} 1 & -1 \ 1 & 1 \end{pmatrix} \begin{pmatrix} 27 & 0 \ 0 & -1 \end{pmatrix} \frac{1}{2}\begin{pmatrix} 1 & 1 \ -1 & 1 \end{pmatrix} = \frac{1}{2} \begin{pmatrix} 27 & 1 \ 27 & -1 \end{pmatrix} \begin{pmatrix} 1 & 1 \ -1 & 1 \end{pmatrix} = \begin{pmatrix} 13 & 14 \ 14 & 13 \end{pmatrix}$.

Common Mistakes

• Order Mismatch: The most frequent error is placing eigenvalues in $D$ in a different order than their corresponding eigenvectors in $P$. If $\lambda_1$ is the first entry in $D$, the first column of $P$ must be the eigenvector associated with $\lambda_1$.
• Ignoring Multiplicity: Assuming a matrix is diagonalisable just because you found all eigenvalues. Always check the linear independence of eigenvectors for repeated eigenvalues.
• Calculation Errors in $P^{-1}$: Ensure you calculate the inverse of $P$ correctly. For $2 \times 2$ matrices, use the formula $\frac{1}{ad-bc}\begin{pmatrix} d & -b \ -c & a \end{pmatrix}$.

Frequently Asked Questions

Q: Can a non-square matrix be diagonalised? No. Diagonalisation is defined only for square $n \times n$ matrices.

Q: What if my eigenvalues are complex? A matrix with complex eigenvalues can still be diagonalised, but the matrix $P$ and $D$ will contain complex numbers.

Q: How do I know if a matrix is defective? A matrix is defective if it does not have enough linearly independent eigenvectors to form a basis for $\mathbb{R}^n$.

Conclusion

Eigenvalue decomposition is a cornerstone of linear algebra that simplifies complex matrix operations into manageable scalar arithmetic. By mastering the relationship between $A$, $P$, and $D$, you gain a powerful tool for solving differential equations, analysing dynamical systems, and performing matrix exponentiation. To see these concepts in motion, visit MathInstructor AI to generate a free, narrated animated lesson on this topic.