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Eigenvalues:
Special Matrices

Discover how triangular, diagonal, and other special matrices reveal their eigenvalues instantly: just read the diagonal.

Linear Algebra

What Are Eigenvalues?

An eigenvalue \(\lambda\) of a square matrix \(A\) is a scalar such that there exists a nonzero vector \(\mathbf{v}\) satisfying:

The vector \(\mathbf{v}\) is called an eigenvector. Eigenvalues tell us how a linear transformation scales vectors in certain directions.

The Characteristic Equation

To find eigenvalues in general, we solve the characteristic equation:

For an \(n \times n\) matrix, this gives a degree-\(n\) polynomial in \(\lambda\). But for special matrices, the determinant simplifies dramatically: no polynomial expansion needed.

What Makes a Matrix "Special"?

Upper Triangular
All entries below the main diagonal are zero. Example: \(\begin{bmatrix} 2 & 3 \\ 0 & 5 \end{bmatrix}\)
Lower Triangular
All entries above the main diagonal are zero. Example: \(\begin{bmatrix} 4 & 0 \\ 1 & 7 \end{bmatrix}\)
Diagonal
All off-diagonal entries are zero. Both upper and lower triangular. Example: \(\begin{bmatrix} 3 & 0 \\ 0 & 9 \end{bmatrix}\)

Visualizing Triangular Structure

Upper Triangular \(a\) \(b\) \(c\) * * * 0 0 0 eigenvalues: \(a, b, c\) Lower Triangular \(d\) \(e\) \(f\) * * * 0 0 0 eigenvalues: \(d, e, f\) Highlighted cells = diagonal entries = eigenvalues

The Core Theorem

If \(A\) is \(n \times n\) and triangular with diagonal entries \(a_{11}, a_{22}, \dots, a_{nn}\), then:

So \(\lambda_1 = a_{11},\; \lambda_2 = a_{22},\; \dots,\; \lambda_n = a_{nn}\).

Why Does This Work?

1
Form \(A - \lambda I\): Subtracting \(\lambda\) from the diagonal of a triangular matrix gives another triangular matrix.
2
Determinant of a triangular matrix: The determinant is the product of its diagonal entries: \(\det(A - \lambda I) = \prod_{i=1}^{n}(a_{ii} - \lambda)\).
3
Set equal to zero: Each factor \((a_{ii} - \lambda) = 0\) gives \(\lambda = a_{ii}\). No expansion needed.

Triangular vs. General Matrices

Triangular Matrix General Matrix
Eigenvalue method Read diagonal entries Solve \(\det(A - \lambda I) = 0\)
Effort Instant Polynomial expansion
Characteristic polynomial Already factored Must factor manually
Repeated eigenvalues? Yes: repeated diagonal entries Yes: repeated roots

How to Apply the Theorem

1
Check structure: Verify the matrix is upper triangular, lower triangular, or diagonal. All entries on one side of the diagonal must be zero.
2
Read the diagonal: List all entries \(a_{11}, a_{22}, \dots, a_{nn}\). These are the eigenvalues.
3
Note multiplicities: If a value appears \(k\) times on the diagonal, it has algebraic multiplicity \(k\).
4
Done: No determinant computation, no polynomial factoring required.

Example 1: Upper Triangular

Find the eigenvalues of:

Step 1: This is upper triangular: all entries below the diagonal are 0.

Step 2: Read the diagonal: \(3\), \(-2\), \(5\).

Example 2: Lower Triangular

Find the eigenvalues of:

Step 1: This is lower triangular: all entries above the diagonal are 0.

Step 2: Diagonal entries: \(7, 7, 4, -1\).

Example 3: Diagonal Matrix

Find the eigenvalues of:

A diagonal matrix is both upper and lower triangular. The eigenvalues are the diagonal entries: \(0, -3, 0\).

Other Special Matrices

Identity \(I_n\)
Diagonal with all 1s. Eigenvalue: \(\lambda = 1\) with multiplicity \(n\).
Zero Matrix
Diagonal with all 0s. Eigenvalue: \(\lambda = 0\) with multiplicity \(n\).
Scalar \(cI_n\)
Diagonal with all \(c\). Eigenvalue: \(\lambda = c\) with multiplicity \(n\).

Useful Properties

1
Trace: The sum of eigenvalues equals the trace (sum of diagonal entries): \(\lambda_1 + \lambda_2 + \cdots + \lambda_n = \text{tr}(A)\).
2
Determinant: The product of eigenvalues equals \(\det(A)\): \(\lambda_1 \cdot \lambda_2 \cdots \lambda_n = \det(A)\).
3
Singularity: \(A\) is singular if and only if at least one eigenvalue is 0.

Common Mistakes

✕ Misidentifying Structure
Thinking a matrix is triangular when it's not. Always check that all entries on one side of the diagonal are zero.
✕ Ignoring Multiplicity
Listing \(\lambda = 7\) once when it appears twice on the diagonal. Repeated entries mean higher algebraic multiplicity.
✕ Forgetting \(\lambda = 0\)
Skipping zero as an eigenvalue. Zero is a valid eigenvalue and signals the matrix is singular.

Summary Table

Matrix Type Eigenvalues Example
Upper Triangular Diagonal entries \(\begin{bmatrix} 2 & 1 \\ 0 & 5 \end{bmatrix}\) → \(2, 5\)
Lower Triangular Diagonal entries \(\begin{bmatrix} 4 & 0 \\ 3 & 1 \end{bmatrix}\) → \(4, 1\)
Diagonal Diagonal entries \(\begin{bmatrix} 6 & 0 \\ 0 & -2 \end{bmatrix}\) → \(6, -2\)
Identity \(I_n\) \(\lambda = 1\) (mult. \(n\)) \(\begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}\) → \(1, 1\)
Zero Matrix \(\lambda = 0\) (mult. \(n\)) \(\begin{bmatrix} 0 & 0 \\ 0 & 0 \end{bmatrix}\) → \(0, 0\)

Question 1 of 5

What are the eigenvalues of \(A = \begin{bmatrix} 4 & 0 & 0 \\ 2 & -1 & 0 \\ 5 & 3 & 6 \end{bmatrix}\)?

Question 2 of 5

A \(3 \times 3\) upper triangular matrix has diagonal entries \(2, 2, 5\). What is the algebraic multiplicity of \(\lambda = 2\)?

Question 3 of 5

If a diagonal matrix has eigenvalues \(3, 0, -2\), which statement is true?

Question 4 of 5

For the matrix \(M = \begin{bmatrix} 1 & 7 & -3 \\ 0 & 1 & 9 \\ 0 & 0 & 1 \end{bmatrix}\), what is \(\det(M)\)?

Question 5 of 5

Which matrix does NOT allow you to read eigenvalues directly from the diagonal?