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Eigenvalues and Eigenvectors Short Notes for GATE CS

Eigenvalues and Eigenvectors short notes for GATE CS: 7 study cards covering concepts, formulas, shortcuts and exam traps, plus solved practice questions.

eigenvalues and eigenvectors short notes

Topic Summary: Determinants and Multiplicity

Topic Summary: Determinants and Multiplicity Core Formulas Product: det (A) = \prod λ_{i} Sum: Tr (A) = \sum λ_{i} Geometric Multiplicity: GM = n - rank (A - λ I) The Golden Inequality 1 \leq GM \leq AM Key Exam Takeaways Maximum algebraic multiplicity in an n \times n matrix is n . Never calculate eigenvalues just to find their product or sum; use determinant and trace. A matrix is defective (not diagonalizable) if GM < AM for any eigenvalue. If rows/columns are linearly dependent, det (A) = 0, meaning at least one eigenvalue is zero.

Verification Quick Checklist

Non-zero rule: v \neq = 0 is mandatory. The Test: Compute A v . The Check: Does A v = c v for some scalar c ? Yes \to Eigenvector (λ = c). No \to Not an eigenvector. Equivalence: v and k v (k \neq = 0) represent the same eigenvector direction and share the same λ . @keyframes checkIn { from { opacity: 0; transform: translateX(-10px); } to { opacity: 1; transform: translateX(0); } }

Formula Matrix

#sc-sn01 .grid { display: grid; grid-template-columns: 1fr 2fr; gap: 1px; background: var(--color-border); border: 1px solid var(--color-border); border-radius: 4px; overflow: hidden; font-size: 0.85rem; } #sc-sn01 .hdr { background: var(--color-bg); padding: 0.5rem; font-weight: 700; color: var(--color-text); } #sc-sn01 .cell { background: var(--color-bg); padding: 0.5rem; line-height: 1.3; } #sc-sn01 .cell.accent { background: color-mix(in srgb, var(--color-primary) 10%, transparent); color: var(--color-primary); font-weight: 600; } Rule Formula & Condition Trace \sum λ_{i} = tr (A) = 0 (Simple graph) Regular λ_{m a x} = k (Connected k -regular) Bipartite λ \in Spec ⟺ - λ \in Spec (Bipartite) Complete K_{n} {n - 1, - 1^{(n - 1)}} (Complete graph) Complete Bip. K_{m, n} {mn, - mn, 0^{(m + n - 2)}} (Complete bipartite) Disjoint Union Spec (G_{1} \cup G_{2}) = Spec (G_{1}) \cup Spec (G_{2}) Sum of Squares \sum λ_{i}^{2} = tr (A^{2}) = 2∣ E ∣ (Simple graph)

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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 Short Notes for GATE CS

Eigenvalues and Eigenvectors short notes for GATE CS: 7 study cards covering concepts, formulas, shortcuts and exam traps, plus solved practice questions.

Topic Summary: Determinants and Multiplicity

Core Formulas

Product: $det (A) = \prod λ_{i}$
Sum: $Tr (A) = \sum λ_{i}$
Geometric Multiplicity: $GM = n - rank (A - λ I)$

The Golden Inequality

1 \leq GM \leq AM

Key Exam Takeaways

Maximum algebraic multiplicity in an $n \times n$ matrix is $n$ .
Never calculate eigenvalues just to find their product or sum; use determinant and trace.
A matrix is defective (not diagonalizable) if $GM < AM$ for any eigenvalue.
If rows/columns are linearly dependent, $det (A) = 0$ , meaning at least one eigenvalue is zero.

Verification Quick Checklist

Non-zero rule: $v \neq = 0$ is mandatory.

The Test: Compute $A v$ .

The Check: Does $A v = c v$ for some scalar $c$ ?

Yes $\to$ Eigenvector ( $λ = c$ ).
No $\to$ Not an eigenvector.

Equivalence: $v$ and $k v$ ( $k \neq = 0$ ) represent the same eigenvector direction and share the same $λ$ .

Formula Matrix

Rule

Formula & Condition

Trace

\sum λ_{i} = tr (A) = 0

(Simple graph)

Regular

λ_{m a x} = k

(Connected $k$ -regular)

Bipartite

λ \in Spec ⟺ - λ \in Spec

(Bipartite)

Complete

K_{n}

{n - 1, - 1^{(n - 1)}}

(Complete graph)

Complete Bip.

K_{m, n}

{mn, - mn, 0^{(m + n - 2)}}

(Complete bipartite)

Disjoint Union

Spec (G_{1} \cup G_{2}) = Spec (G_{1}) \cup Spec (G_{2})

Sum of Squares

\sum λ_{i}^{2} = tr (A^{2}) = 2∣ E ∣

(Simple graph)

If You See This, Do This

Stem Trigger

First Move

"sum of eigenvalues"

Check trace. Simple graph

\to 0

"sum of squares"

Calculate

2 \times ∣ E ∣

k

-regular" + "largest"

Answer is

k

"bipartite" + "

λ

Write

- λ

immediately.

"complete

K_{n}

Write

n - 1

and

(n - 1)

copies of

- 1

"complete bip.

K_{m, n}

Write

\pm mn

and

(m + n - 2)

zeros.

"disjoint union"

Concatenate individual spectra.

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 short notes

Topic Summary: Determinants and Multiplicity

Topic Summary: Determinants and Multiplicity

Verification Quick Checklist

Formula Matrix

Try a question

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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 Short Notes for GATE CS

Topic Summary: Determinants and Multiplicity

Topic Summary: Determinants and Multiplicity

Verification Quick Checklist

Formula Matrix

If You See This, Do This

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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