Engineering Mathematics 10 | Linear Algebra : Eigen Values & Eigen Vector Decoded- Episode 3 | GATE

Main Ideas

The session reviews eigenvalue and eigenvector fundamentals, then applies them to exam-style questions. A recurring theme is to choose the quickest reliable method: use known matrix properties or test answer choices when possible, and use the standard eigenvector equations when necessary.

Key Properties Reviewed

• Rank: The rank of a matrix equals the number of linearly independent rows and also the number of linearly independent columns.
• Null space: For a homogeneous system (Ax=0), the nullity gives the number of free variables and the dimension of the solution space.

• Eigenvectors and nullity: For an eigenvalue (\lambda), eigenvectors satisfy [ (A-\lambda I)x=0. ] The nullity of (A-\lambda I) is the number of linearly independent eigenvectors associated with (\lambda), also called its geometric multiplicity.

• Repeated eigenvalues: If an eigenvalue is repeated, its number of independent eigenvectors cannot exceed its multiplicity.

• Distinct eigenvalues: Eigenvectors corresponding to distinct eigenvalues are linearly independent.
• Symmetric matrices: Eigenvectors corresponding to distinct eigenvalues of a real symmetric matrix are orthogonal.

• Trace and determinant: The trace is the sum of the eigenvalues, and the determinant is their product. For a (2\times2) matrix with eigenvalues (\lambda_1,\lambda_2), [ \lambda_1+\lambda_2=\operatorname{tr}(A),\qquad \lambda_1\lambda_2=\det(A). ] Also, [ \lambda_1^2+\lambda_2^2 =(\lambda_1+\lambda_2)^2-2\lambda_1\lambda_2, ] which can help recover their product when the trace and the sum of squared eigenvalues are known.

• Non-singular matrices: A non-singular matrix cannot have zero as an eigenvalue.

• Eigenvalue transformations: Matrix expressions built from (A), such as powers of (A), can often be handled by applying the corresponding expression to its eigenvalues.

Methods for Solving Questions

• Use the characteristic equation when needed. Compute [ \det(A-\lambda I)=0 ] to find eigenvalues. When simplifying the determinant, use valid row or column operations.

• Find eigenvectors by solving a homogeneous system. Once (\lambda) is known, solve ((A-\lambda I)x=0). Row-reducing can make the relationships between the components clear.

• Remember that eigenvectors are not unique in scale. Any nonzero scalar multiple of an eigenvector is also an eigenvector. When matching a vector given in a question, compare component ratios rather than assuming a particular normalization.

• Exploit a supplied eigenvector. If the question gives a matrix and a vector (x), test or use [ Ax=\lambda x. ] This can be quicker than independently finding all eigenvectors.

• Test answer choices directly. In multiple-choice questions, substitute a proposed vector into (Ax=\lambda x). Compare the resulting components to check whether the same (\lambda) works throughout. This can eliminate options without solving the full problem.

• Use trace and determinant to check results. Once eigenvalues have been found, their sum should match the trace and their product should match the determinant.
• Recognize block structure. For block-diagonal matrices—and the related block-triangular forms discussed in the lecture—the eigenvalues come from the diagonal blocks. Find the eigenvalues of each smaller block separately rather than expanding a large characteristic determinant. The determinant is the product of the determinants of the blocks.
• Use determinant sign rules. Interchanging two rows or columns changes the determinant’s sign. This can establish relationships between determinants without calculating each one in full.

Exam and Study Advice

The instructor recommends practicing previous-year questions and developing a habit of spotting shortcuts, especially in objective questions. The lecture also encourages students to focus on the material currently being taught rather than becoming distracted by questions about later topics or course completion. Further linear algebra topics are previewed for a subsequent session.

Speakers and Sources Featured

• Instructor: An unnamed GATE Wallah mathematics instructor, explaining the concepts and worked examples.
• Students/viewers: Participants appear through live-chat answers and questions; no individual student is clearly identifiable as a sustained speaker.