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Eigenvalues and Eigenvectors Notes for GATE CS

Eigenvalues and Eigenvectors notes for GATE CS: 35 study cards covering concepts, formulas, shortcuts and exam traps, plus solved practice questions.

eigenvalues and eigenvectors notes

Linear Algebra: Eigenvalues and Eigenvectors Roadmap

Linear Algebra: Eigenvalues and Eigenvectors Roadmap 1. Eigenvalues, Determinants and Multiplicity The foundation: characteristic equations, trace, determinant, and root multiplicity. Current Topic 2. Eigenvector Verification Test vectors and compute null spaces to verify eigenvectors. 3. Spectra of Graph Adjacency Matrices Apply eigenvalues to graph theory and symmetric matrices. 4. Eigenvalues of Matrix Powers Predict long-term behavior and compute high matrix powers efficiently.

The Core Idea: Eigenvalues, Determinants, and Multiplicity

The Core Idea: Eigenvalues, Determinants, and Multiplicity v Av = λv @keyframes stretch { 0% { x2: 60; y2: 40; } 100% { x2: 130; y2: -30; } } When a matrix multiplies a vector, it usually changes both the length and the direction of that vector. However, for any square matrix, there are special directions that only stretch or shrink, without rotating. Eigenvectors are the vectors that point in these special directions. Eigenvalues are the pure scaling factors (how much they stretch or shrink). The determinant is the overall volume scaling factor of the entire space. Multiplicity tells us how many independent directions share the exact same scaling factor.

The Characteristic Equation

The Characteristic Equation To find the eigenvalues mathematically, we look for vectors v that satisfy: A v = λ v Rearranging this gives: (A - λ I) v = 0 For a non-zero eigenvector v to exist, the matrix (A - λ I) must be singular (non-invertible). This means its determinant must be zero: ∣ A - λ I ∣ = 0 This is the characteristic equation . Expanding this determinant yields a polynomial in λ of degree n (for an n \times n matrix). The roots of this polynomial are the eigenvalues.

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Question 1

Level 1: Warm-up

The eigenvalues of a 3 \times 3 matrix are 2, - 3, and 4 . What is the determinant of the matrix?

Question 2

Level 1: Warm-up

Let A = [4007] and v = [10] . If A v = λ v, what is the value of λ ?

Question 3

Level 1: Warm-up

For an n \times n matrix, the characteristic equation ∣ A - λ I ∣ = 0 yields a polynomial in λ . What is the maximum possible degree of this polynomial for a 5 \times 5 matrix?

Question 4

Level 1: Warm-up

The trace of a 3 \times 3 matrix is 12. If two of its eigenvalues are 4 and 5, consider the following statement: "The third eigenvalue is 3." Is this statement true or false?

Question 5

Level 1: Warm-up

Let A be a 3 \times 3 matrix where the third row is exactly the sum of the first two rows. Which of the following must be an eigenvalue of A ?

Question 6

Level 1: Warm-up

A linear transformation scales the vector v by a factor of - 5 and reverses its direction. What is the eigenvalue associated with v ?

Question 7

Level 1: Warm-up

Consider the characteristic equation of a 4 \times 4 matrix: (λ - 3)^{2} (λ + 2)^{2} = 0 . Assertion (A): The algebraic multiplicity of λ = 3 is 2. Reason (R): The geometric multiplicity of λ = 3 is 4.

Question 8

Level 1: Warm-up

The characteristic equation of a matrix has roots λ_{1} = - 3 and λ_{2} = 2 . The algebraic multiplicity of λ_{1} is 3, and for λ_{2} it is 1. What is the sum of the maximum possible geometric multiplicities for all eigenvalues?

Question 9

Level 1: Warm-up

Consider the following statements regarding eigenvector verification for a large 4 \times 4 matrix: Assertion (A): To verify if a given vector is an eigenvector, one should first compute the characteristic equation det (A - λ I) = 0 . Reason (R): Matrix-vector multiplication is computationally cheaper than expanding a 4 \times 4 determinant.

Question 10

Level 1: Warm-up

By definition, an eigenvector of a square matrix must satisfy a strict non-zero condition. What is the maximum number of zero vectors that can be included in a valid set of eigenvectors for any matrix?

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Eigenvalues and Eigenvectors Notes for GATE CS

Eigenvalues and Eigenvectors notes for GATE CS: 35 study cards covering concepts, formulas, shortcuts and exam traps, plus solved practice questions.

Linear Algebra: Eigenvalues and Eigenvectors Roadmap

1. Eigenvalues, Determinants and Multiplicity

The foundation: characteristic equations, trace, determinant, and root multiplicity. Current Topic

2. Eigenvector Verification

Test vectors and compute null spaces to verify eigenvectors.

3. Spectra of Graph Adjacency Matrices

Apply eigenvalues to graph theory and symmetric matrices.

4. Eigenvalues of Matrix Powers

Predict long-term behavior and compute high matrix powers efficiently.

The Core Idea: Eigenvalues, Determinants, and Multiplicity

When a matrix multiplies a vector, it usually changes both the length and the direction of that vector.

However, for any square matrix, there are special directions that only stretch or shrink, without rotating.

Eigenvectors are the vectors that point in these special directions.
Eigenvalues are the pure scaling factors (how much they stretch or shrink).

The determinant is the overall volume scaling factor of the entire space. Multiplicity tells us how many independent directions share the exact same scaling factor.

The Characteristic Equation

To find the eigenvalues mathematically, we look for vectors $v$ that satisfy:

A v = λ v

Rearranging this gives:

(A - λ I) v = 0

For a non-zero eigenvector $v$ to exist, the matrix $(A - λ I)$ must be singular (non-invertible). This means its determinant must be zero:

∣ A - λ I ∣ = 0

This is the characteristic equation. Expanding this determinant yields a polynomial in $λ$ of degree $n$ (for an $n \times n$ matrix). The roots of this polynomial are the eigenvalues.

What is Multiplicity?

When you solve the characteristic equation $∣ A - λ I ∣ = 0$ , you might get repeated roots. For example, the polynomial might be $(λ - 2)^{3} (λ - 5) = 0$ .

Here, the eigenvalue $λ = 2$ repeats three times. This repetition is called multiplicity.

However, in linear algebra, multiplicity is not just a single number. We must strictly distinguish between two different types:

Algebraic Multiplicity (AM)
Geometric Multiplicity (GM)

Confusing these two is one of the most common traps in competitive exams.

Eigenvalues and Eigenvectors: Solved Questions with Step-by-Step Explanations (10 Problems)

Question 1 · Engineering Mathematics MCQ

The eigenvalues of a $3 \times 3$ matrix are $2$ , $- 3$ , and $4$ . What is the determinant of the matrix?

A.
-24
B.
24
C.
-1
D.
3

Correct Answer:

Step-by-Step Solution

Key idea: The determinant of a matrix is exactly equal to the product of all its eigenvalues.

Step 1: Identify the eigenvalues: $λ_{1} = 2$ , $λ_{2} = - 3$ , $λ_{3} = 4$ .

Step 2: Multiply them together: $det (A) = 2 \times (- 3) \times 4$ .

Step 3: Calculate the product: $- 6 \times 4 = - 24$ .

Answer: -24

Question 2 · Engineering Mathematics MCQ

Let $A = [4007]$ and $v = [10]$ . If $A v = λ v$ , what is the value of $λ$ ?

A.
4
B.
7
C.
11
D.
28

Correct Answer:

Step-by-Step Solution

Key idea: This is a direct substitution question for the eigenvalue equation $A v = λ v$ .

Step 1: Multiply matrix $A$ by vector $v$ .

$A v = [4007] [10] = [40]$

Step 2: Set this equal to $λ v$ .

$[40] = λ [10] = [λ 0]$

Step 3: Compare the components to find $λ = 4$ .

Answer: 4

Question 3 · Engineering Mathematics MCQ

For an $n \times n$ matrix, the characteristic equation $∣ A - λ I ∣ = 0$ yields a polynomial in $λ$ . What is the maximum possible degree of this polynomial for a $5 \times 5$ matrix?

A.
5
B.
10
C.
25
D.
125

Correct Answer:

Step-by-Step Solution

Key idea: The degree of the characteristic polynomial is always exactly equal to the size of the square matrix.

Step 1: The matrix is of size $5 \times 5$ , so $n = 5$ .

Step 2: The determinant $∣ A - λ I ∣$ expands to a polynomial of degree $n$ .

Step 3: Therefore, the degree is 5.

Answer: 5

Question 4 · Engineering Mathematics MCQ

The trace of a $3 \times 3$ matrix is 12. If two of its eigenvalues are 4 and 5, consider the following statement: "The third eigenvalue is 3." Is this statement true or false?

A.
True, because the sum of eigenvalues equals the trace
B.
True, because the product of eigenvalues equals the trace
C.
False, because the third eigenvalue is 21
D.
False, because the third eigenvalue is 60

Correct Answer:

Step-by-Step Solution

Key idea: The trace of a matrix is equal to the sum of its eigenvalues.

Step 1: Let the third eigenvalue be $λ_{3}$ . The trace is the sum of eigenvalues: $4 + 5 + λ_{3} = 12$ .

Step 2: Solve for $λ_{3}$ : $9 + λ_{3} = 12 ⟹ λ_{3} = 3$ .

Step 3: The statement is true, and the reason is that the sum of eigenvalues equals the trace.

Answer: True, because the sum of eigenvalues equals the trace.

Question 5 · Engineering Mathematics MCQ

Let $A$ be a $3 \times 3$ matrix where the third row is exactly the sum of the first two rows. Which of the following must be an eigenvalue of $A$ ?

A.
0
B.
1
C.
3
D.
-1

Correct Answer:

Step-by-Step Solution

Key idea: This is a singular matrix recognition question. Linearly dependent rows imply a zero determinant, which forces at least one eigenvalue to be zero.

Step 1: The third row is the sum of the first two rows, meaning the rows of $A$ are linearly dependent.

Step 2: A matrix with linearly dependent rows is singular, which means its determinant is exactly zero ( $det (A) = 0$ ).

Step 3: The product of all eigenvalues equals the determinant. Since $det (A) = 0$ , at least one eigenvalue must be $0$ .

Answer: 0

Question 6 · Engineering Mathematics MCQ

A linear transformation scales the vector $v$ by a factor of $- 5$ and reverses its direction. What is the eigenvalue associated with $v$ ?

A.
5
B.
-5
C.
0.2
D.
-0.2

Correct Answer:

Step-by-Step Solution

Key idea: The eigenvalue $λ$ represents the pure scaling factor, including any sign change for direction reversal.

Step 1: The problem states the scaling factor is $- 5$ and the direction is reversed.

Step 2: Reversing direction means the scalar is negative. The magnitude of scaling is 5.

Step 3: Therefore, the eigenvalue is $λ = - 5$ .

Answer: -5

Question 7 · Engineering Mathematics MCQ

Consider the characteristic equation of a $4 \times 4$ matrix: $(λ - 3)^{2} (λ + 2)^{2} = 0$ .

Assertion (A): The algebraic multiplicity of $λ = 3$ is 2.

Reason (R): The geometric multiplicity of $λ = 3$ is 4.

A.
Both A and R are true and R is the correct explanation of A
B.
Both A and R are true but R is NOT the correct explanation of A
C.
A is true but R is false
D.
A is false but R is true

Correct Answer:

Step-by-Step Solution

Key idea: Algebraic multiplicity is the power of the factor in the characteristic equation, while geometric multiplicity is the number of independent eigenvectors (bounded by algebraic multiplicity).

Step 1: The factor $(λ - 3)$ is squared, so the algebraic multiplicity of $λ = 3$ is 2. Assertion (A) is true.

Step 2: The geometric multiplicity must be between 1 and the algebraic multiplicity (2). It cannot be 4. Reason (R) is false.

Step 3: Therefore, A is true but R is false.

Answer: A is true but R is false.

Question 8 · Engineering Mathematics MCQ

The characteristic equation of a matrix has roots $λ_{1} = - 3$ and $λ_{2} = 2$ . The algebraic multiplicity of $λ_{1}$ is 3, and for $λ_{2}$ it is 1. What is the sum of the maximum possible geometric multiplicities for all eigenvalues?

A.
3
B.
4
C.
5
D.
6

Correct Answer:

Step-by-Step Solution

Key idea: The geometric multiplicity (GM) of an eigenvalue is bounded by its algebraic multiplicity (AM). The maximum possible GM is exactly equal to the AM.

Step 1: For $λ_{1} = - 3$ , the algebraic multiplicity is 3. The maximum possible geometric multiplicity is therefore 3.

Step 2: For $λ_{2} = 2$ , the algebraic multiplicity is 1. The maximum possible geometric multiplicity is 1.

Step 3: The sum of the maximum possible geometric multiplicities is $3 + 1 = 4$ .

Answer: 4

Question 9 · Engineering Mathematics MCQ

Consider the following statements regarding eigenvector verification for a large $4 \times 4$ matrix:

Assertion (A): To verify if a given vector is an eigenvector, one should first compute the characteristic equation $det (A - λ I) = 0$ .

Reason (R): Matrix-vector multiplication is computationally cheaper than expanding a $4 \times 4$ determinant.

A.
Both A and R are true and R is the correct explanation of A
B.
Both A and R are true but R is NOT the correct explanation of A
C.
A is false but R is true
D.
A is true but R is false

Correct Answer:

Step-by-Step Solution

Key idea: When candidate vectors are provided, direct verification via multiplication is vastly superior to solving the characteristic equation.

Step 1: Evaluate Assertion (A): Computing the characteristic equation for a $4 \times 4$ matrix is computationally heavy and unnecessary if you only need to verify specific candidate vectors. Thus, A is false.

Step 2: Evaluate Reason (R): Matrix-vector multiplication requires $O (n^{2})$ operations, while expanding a $4 \times 4$ determinant is much more complex and prone to error. Thus, R is true.

Step 3: Since A is false and R is true, the correct option is the third one.

Answer: A is false but R is true.

Question 10 · Engineering Mathematics MCQ

By definition, an eigenvector of a square matrix must satisfy a strict non-zero condition. What is the maximum number of zero vectors that can be included in a valid set of eigenvectors for any matrix?

A.
0
B.
1
C.
2
D.
Depends on the matrix size

Correct Answer:

Step-by-Step Solution

Key idea: The definition of an eigenvector explicitly excludes the zero vector.

Step 1: An eigenvector $v$ of a matrix $A$ must satisfy $A v = λ v$ .

Step 2: While the zero vector trivially satisfies $A 0 = λ 0$ for any $λ$ , it provides no information about directional scaling.

Step 3: Therefore, the strict definition requires $v \neq = 0$ . The zero vector can never be an eigenvector.

Answer: 0

eigenvalues and eigenvectors notes

Linear Algebra: Eigenvalues and Eigenvectors Roadmap

Linear Algebra: Eigenvalues and Eigenvectors Roadmap

The Core Idea: Eigenvalues, Determinants, and Multiplicity

The Core Idea: Eigenvalues, Determinants, and Multiplicity

The Characteristic Equation

The Characteristic Equation

Try a question

Login to view the complete notes

Personalised first. High quality throughout.

Built around you, not around a syllabus PDF

Revision that hits your weak spots

Questions calibrated to the real exam

Notes written for recall, not for volume

One place for everything

Honest progress

Unlock the whole course

Eigenvalues and Eigenvectors Notes for GATE CS

Linear Algebra: Eigenvalues and Eigenvectors Roadmap

Linear Algebra: Eigenvalues and Eigenvectors Roadmap

The Core Idea: Eigenvalues, Determinants, and Multiplicity

The Core Idea: Eigenvalues, Determinants, and Multiplicity

The Characteristic Equation

The Characteristic Equation

What is Multiplicity?

What is Multiplicity?

Eigenvalues and Eigenvectors: Solved Questions with Step-by-Step Explanations (10 Problems)

Step-by-Step Solution

Step-by-Step Solution

Step-by-Step Solution

Step-by-Step Solution

Step-by-Step Solution

Step-by-Step Solution

Step-by-Step Solution

Step-by-Step Solution

Step-by-Step Solution

Step-by-Step Solution

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