Eigenvalues and Eigenvectors

The defining equation

For a square matrix $A$ , an eigenvector is a nonzero vector $v$ that $A$ only stretches or shrinks along its own line: multiplying by $A$ is the same as multiplying by a number $\lambda$ , the eigenvalue:

Av = \lambda v \quad\Longleftrightarrow\quad (A-\lambda I)v = 0

Since $v \ne 0$ , this has a solution only when $A-\lambda I$ is singular, which gives the characteristic equation:

\det(A - \lambda I) = 0

Worked example

Find the eigenvalues and eigenvectors of $A = \begin{bmatrix} 2 & 1 \\ 1 & 2 \end{bmatrix}$ .

\det\begin{bmatrix} 2-\lambda & 1 \\ 1 & 2-\lambda \end{bmatrix} = (2-\lambda)^2 - 1 = 0 \;\Rightarrow\; \lambda = 1 \text{ or } \lambda = 3

For $\lambda=1$ : $(A-I)v=0$ gives $v_1+v_2=0$ , so $v = \begin{bmatrix}1\\-1\end{bmatrix}$ . For $\lambda=3$ : $(A-3I)v=0$ gives $v_1=v_2$ , so $v = \begin{bmatrix}1\\1\end{bmatrix}$ .

Check by substitution: $A\begin{bmatrix}1\\-1\end{bmatrix} = \begin{bmatrix}1\\-1\end{bmatrix}$ and $A\begin{bmatrix}1\\1\end{bmatrix} = \begin{bmatrix}3\\3\end{bmatrix}$ . Check by trace and determinant: $1+3 = 4 = 2+2$ and $1\times3 = 3 = 4-1$ .

Worked example: a non-symmetric matrix

Find the eigenvalues and eigenvectors of $B = \begin{bmatrix} 4 & 1 \\ 2 & 3 \end{bmatrix}$ .

\det(B-\lambda I) = (4-\lambda)(3-\lambda) - 2 = \lambda^2 - 7\lambda + 10 = (\lambda-2)(\lambda-5)

For $\lambda=5$ : $-v_1+v_2=0$ , so $v = \begin{bmatrix}1\\1\end{bmatrix}$ . For $\lambda=2$ : $2v_1+v_2=0$ , so $v = \begin{bmatrix}1\\-2\end{bmatrix}$ . Checks: $2+5 = 7 = \text{trace}(B)$ and $2\times5 = 10 = \det B$ .

Shortcuts worth knowing

If A has eigenvalue λ (with the same eigenvector)

Matrix	Eigenvalue	For B above (λ = 2, 5)
$A^2$	$\lambda^2$	4, 25
$A^{-1}$ (if $A$ is invertible)	$1/\lambda$	1/2, 1/5
$A + kI$	$\lambda + k$	with $k = 3$ : 5, 8
$kA$	$k\lambda$	with $k = 2$ : 4, 10

For a triangular (or diagonal) matrix the eigenvalues are the diagonal entries: $\begin{bmatrix} 2 & 5 & -1 \\ 0 & 3 & 4 \\ 0 & 0 & -1 \end{bmatrix}$ has eigenvalues 2, 3 and −1. A real symmetric matrix always has real eigenvalues.

Common mistakes

Writing $\det(A) - \lambda = 0$ . Subtract $\lambda$ from every diagonal entry first, then take the determinant.
Accepting $v = 0$ as an eigenvector. Eigenvectors are nonzero by definition; any nonzero multiple of a valid one also works.
Skipping the checks. The sum of eigenvalues must equal the trace and their product the determinant. If either fails, the characteristic equation has a slip.
Assuming eigenvalues of $A+B$ are sums of eigenvalues. That holds for $A+kI$ , not for two general matrices.

Where this comes up in exams

Eigenvalues and eigenvectors are listed under Linear Algebra in the GATE 2027 syllabi for CS, EC, EE, IN and ME (the EC syllabus writes 'eigen values and eigen vectors').

Checked against GATE 2027 CS syllabus, GATE 2027 EC syllabus, GATE 2027 EE syllabus, GATE 2027 IN syllabus and GATE 2027 ME syllabus. Syllabi can change from year to year, so confirm with the latest official notification for your exam.

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