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Orthogonal, Projection and Special Matrices Notes for GATE DA

Orthogonal, Projection and Special Matrices notes for GATE DA: 18 study cards covering concepts, formulas, shortcuts and exam traps, plus solved practice ques

orthogonal projection and special matrices notes

Chapter Roadmap: Orthogonal, Projection and Special Matrices

Chapter Roadmap Welcome to the geometry of linear transformations. This chapter bridges the gap between abstract matrix algebra and visual geometric intuition. Your Learning Journey 1. Projection Matrices and Quadratic Forms Classifying surfaces, Rayleigh quotient, centering matrix. High Weightage 2. Orthogonal and Involutory Matrices Length-preserving transformations, rotations, own inverses. Moderate Weightage By the end of this chapter, you will: Instantly classify the shape of any quadratic form using eigenvalues. Recognize projection matrices and deduce their eigenvalues without calculation. Solve constrained optimization problems in seconds.

The Geometry of Quadratic Forms

The Geometry of Quadratic Forms A quadratic form is a scalar-valued polynomial where every term has a degree of exactly two. For a column vector x \in R^{n} and an n \times n matrix A, the quadratic form is written as: Q (x) = x^{T} A x Geometric Intuition If you set Q (x) = c for some constant c, you get a geometric surface: In 2D: Conic sections (ellipses, hyperbolas, parabolas). In 3D: Quadric surfaces (ellipsoids, hyperboloids). The matrix A acts as the "DNA" of this surface, dictating its orientation, stretching, and curvature.

The Symmetric Matrix Representation

The Symmetric Matrix Representation Every quadratic form can be represented by a unique symmetric matrix. If you are given a non-symmetric matrix B, the quadratic form x^{T} B x is exactly equal to x^{T} A x, where A is the symmetrized version of B : A = \frac{B + B ^{T}}{2} Why does this work? Because x^{T} B x is a scalar, it equals its own transpose: x^{T} B x = (x^{T} B x)^{T} = x^{T} B^{T} x Averaging the two gives the symmetric equivalent. Golden Rule: Always symmetrize the matrix before analyzing eigenvalues or definiteness.

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Question 1

Level 1: Warm-up

For the centering matrix M = I_{n} - \frac{1}{n} 1 1^{T}, what is the value of M 1 ?

Question 2

Level 1: Warm-up

Let A be a symmetric matrix with eigenvalues 3, 3, and 5 . What is the minimum value of x^{T} A x subject to x^{T} x = 1 ?

Question 3

Level 1: Warm-up

Let A be a 3 \times 3 real symmetric matrix such that the eigenvalues of A^{2} are 1, 4, and 9 . What is the minimum value of x^{T} A x subject to x^{T} x = 1 ?

Question 4

Level 1: Warm-up

Let A be a 3 \times 3 real symmetric matrix such that A^{2} - 7 A + 10 I = 0 . If the trace of A is 12, what is the minimum value of x^{T} A x subject to x^{T} x = 1 ?

Question 5

Level 1: Warm-up

If the eigenvalues of a symmetric matrix A are 2 and - 3, the quadratic form x^{T} A x is classified as:

Question 6

Level 1: Warm-up

A 3 \times 3 matrix P is an orthogonal projection matrix. If the rank of P is 2, what is the sum of its eigenvalues?

Question 7

Level 1: Warm-up

Consider the following statements: Assertion (A): The quadratic form Q (x) = x^{T} B x with B = (1021) is positive definite. Reason (R): The eigenvalues of B are both positive. Which of the following is correct?

Question 8

Level 1: Warm-up

Match the following matrix properties with their correct values for a 3 \times 3 centering matrix M = I_{3} - \frac{1}{3} 1 1^{T} . List I: 1. Trace of M 2. Determinant of M 3. Eigenvalues of M List II: P. 0 and 1 (with multiplicities 2 and 1) Q. 0 R. 2

Question 9

Level 1: Warm-up

Let A = (a 2 2 a) . If a \in {1, 2, 3, 4, 5}, for how many values of a is the expression x^{T} A x positive definite for all x \neq = 0 ?

Question 10

Level 1: Warm-up

Consider the following statements: Assertion (A): The expression Q (x) = x^{T} B x with B = (2052) is positive definite. Reason (R): The diagonal elements of B are positive. Which of the following is correct?

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Orthogonal, Projection and Special Matrices Notes for GATE DA

Orthogonal, Projection and Special Matrices notes for GATE DA: 18 study cards covering concepts, formulas, shortcuts and exam traps, plus solved practice questions.

Chapter Roadmap: Orthogonal, Projection and Special Matrices

Chapter Roadmap

Welcome to the geometry of linear transformations. This chapter bridges the gap between abstract matrix algebra and visual geometric intuition.

Your Learning Journey

1. Projection Matrices and Quadratic Forms

Classifying surfaces, Rayleigh quotient, centering matrix. High Weightage

2. Orthogonal and Involutory Matrices

Length-preserving transformations, rotations, own inverses. Moderate Weightage

By the end of this chapter, you will:

Instantly classify the shape of any quadratic form using eigenvalues.
Recognize projection matrices and deduce their eigenvalues without calculation.
Solve constrained optimization problems in seconds.

The Geometry of Quadratic Forms

A quadratic form is a scalar-valued polynomial where every term has a degree of exactly two.

For a column vector $x \in R^{n}$ and an $n \times n$ matrix $A$ , the quadratic form is written as:

Q (x) = x^{T} A x

Geometric Intuition

If you set $Q (x) = c$ for some constant $c$ , you get a geometric surface:

In 2D: Conic sections (ellipses, hyperbolas, parabolas).
In 3D: Quadric surfaces (ellipsoids, hyperboloids).

The matrix $A$ acts as the "DNA" of this surface, dictating its orientation, stretching, and curvature.

The Symmetric Matrix Representation

Every quadratic form can be represented by a unique symmetric matrix.

If you are given a non-symmetric matrix $B$ , the quadratic form $x^{T} B x$ is exactly equal to $x^{T} A x$ , where $A$ is the symmetrized version of $B$ :

A = \frac{B + B ^{T}}{2}

Why does this work?

Because $x^{T} B x$ is a scalar, it equals its own transpose:

x^{T} B x = (x^{T} B x)^{T} = x^{T} B^{T} x

Averaging the two gives the symmetric equivalent.

Golden Rule: Always symmetrize the matrix before analyzing eigenvalues or definiteness.

Classifying Quadratic Forms

We classify quadratic forms by observing the sign of $Q (x) = x^{T} A x$ for all non-zero vectors $x$ . This classification depends entirely on the eigenvalues of the symmetric matrix $A$ .

Classification	Eigenvalues of $A$	Shape (2D)
Positive Definite	All $λ_{i} > 0$	Ellipse
Negative Definite	All $λ_{i} < 0$	Ellipse (imaginary)
Positive Semi-Definite	All $λ_{i} \geq 0$	Line or point
Negative Semi-Definite	All $λ_{i} \leq 0$	Line or point
Indefinite	Mixed signs ( $+, -$ )	Hyperbola

Exam Shortcut: To check definiteness, just find the eigenvalues of the symmetrized matrix and check their signs.

Orthogonal, Projection and Special Matrices: Solved Questions with Step-by-Step Explanations (10 Problems)

Question 1 · Linear Algebra NAT

For the centering matrix $M = I_{n} - \frac{1}{n} 1 1^{T}$ , what is the value of $M 1$ ?

Correct Answer:

Step-by-Step Solution

Key idea: This is a direct substitution question testing the action of the centering matrix on the all-ones vector.

Step 1: Write down the expression for $M 1$ :

$M 1 = (I_{n} - \frac{1}{n} 1 1^{T}) 1$

Step 2: Distribute the vector $1$ :

$M 1 = I_{n} 1 - \frac{1}{n} 1 (1^{T} 1)$

Step 3: Simplify the terms:

$I_{n} 1 = 1$ .
$1^{T} 1$ is the dot product of the all-ones vector with itself, which equals $n$ (the sum of $n$ ones).

Step 4: Substitute back:

$M 1 = 1 - \frac{1}{n} 1 (n) = 1 - 1 = 0$

Step 5: The result is the zero vector. In NAT format asking for "the value" or implying a magnitude/component context where 0 is the unique numeric answer, the answer is 0.

Answer: 0

Question 2 · Linear Algebra MCQ

Let $A$ be a symmetric matrix with eigenvalues $3, 3,$ and $5$ . What is the minimum value of $x^{T} A x$ subject to $x^{T} x = 1$ ?

A.
3
B.
5
C.
9
D.
15

Correct Answer:

Step-by-Step Solution

Key idea: The minimum value of a quadratic form $x^{T} A x$ on the unit sphere is the smallest eigenvalue of $A$ .

Step 1: The matrix $A$ is symmetric with eigenvalues $3, 3,$ and $5$ .

Step 2: By the Rayleigh quotient theorem, the minimum value is $λ_{m i n}$ .

Step 3: The smallest eigenvalue is $3$ .

Answer: 3

Question 3 · Linear Algebra MCQ

Let $A$ be a $3 \times 3$ real symmetric matrix such that the eigenvalues of $A^{2}$ are $1, 4,$ and $9$ . What is the minimum value of $x^{T} A x$ subject to $x^{T} x = 1$ ?

A.
-9
B.
-3
C.
1
D.
3

Correct Answer:

Step-by-Step Solution

Key idea: The minimum value of a quadratic form on the unit sphere is the smallest eigenvalue of the symmetric matrix.

Step 1: If the eigenvalues of $A^{2}$ are $1, 4, 9$ , then the eigenvalues of $A$ must be $\pm 1, \pm 2, \pm 3$ .

Step 2: Since $A$ is a real symmetric matrix, its eigenvalues are real. The possible sets of eigenvalues for $A$ are formed by choosing one sign for each magnitude.

Step 3: To find the absolute minimum possible value of the quadratic form, we want the most negative eigenvalue possible for $A$ .

Step 4: The most negative choice is $- 3$ . Thus, the minimum value of $x^{T} A x$ is $- 3$ .

Answer: -3

Question 4 · Linear Algebra MCQ

Let $A$ be a $3 \times 3$ real symmetric matrix such that $A^{2} - 7 A + 10 I = 0$ . If the trace of $A$ is $12$ , what is the minimum value of $x^{T} A x$ subject to $x^{T} x = 1$ ?

A.
2
B.
5
C.
12
D.
10

Correct Answer:

Step-by-Step Solution

Key idea: The minimum value of a quadratic form on the unit sphere is the smallest eigenvalue of the symmetric matrix.

Step 1: The matrix equation $A^{2} - 7 A + 10 I = 0$ implies that the minimal polynomial of $A$ divides $λ^{2} - 7 λ + 10 = 0$ .

Step 2: The roots of $λ^{2} - 7 λ + 10 = 0$ are $λ = 2$ and $λ = 5$ . Thus, the eigenvalues of $A$ can only be $2$ or $5$ .

Step 3: Since $A$ is a $3 \times 3$ matrix, it has exactly 3 eigenvalues. Let $k$ be the number of eigenvalues equal to $5$ , and $3 - k$ be the number equal to $2$ .

Step 4: The trace of $A$ is the sum of its eigenvalues: $5 k + 2 (3 - k) = 12$ .

Step 5: Solving for $k$ : $5 k + 6 - 2 k = 12 ⟹ 3 k = 6 ⟹ k = 2$ .

Step 6: The eigenvalues of $A$ are $5, 5,$ and $2$ .

Step 7: The minimum value of $x^{T} A x$ on the unit sphere is $λ_{m i n} = 2$ .

Answer: 2

Question 5 · Linear Algebra MCQ

If the eigenvalues of a symmetric matrix $A$ are $2$ and $- 3$ , the quadratic form $x^{T} A x$ is classified as:

A.
Positive Definite
B.
Negative Definite
C.
Indefinite
D.
Positive Semi-Definite

Correct Answer:

Step-by-Step Solution

Key idea: This is a classification question based on the signs of eigenvalues.

Step 1: Recall the classification rules for quadratic forms using eigenvalues ( $λ$ ):

Positive Definite: All $λ > 0$ .
Negative Definite: All $λ < 0$ .
Indefinite: Some $λ > 0$ and some $λ < 0$ .

Step 2: Examine the given eigenvalues: $2$ (positive) and $- 3$ (negative).

Step 3: Since there is a mix of positive and negative eigenvalues, the form is Indefinite.

Answer: C

Question 6 · Linear Algebra MCQ

A $3 \times 3$ matrix $P$ is an orthogonal projection matrix. If the rank of $P$ is $2$ , what is the sum of its eigenvalues?

A.
1
B.
2
C.
3
D.
0

Correct Answer:

Step-by-Step Solution

Key idea: The eigenvalues of an orthogonal projection matrix are strictly $0$ and $1$ .

Step 1: The rank of a projection matrix equals the number of eigenvalues that are $1$ .

Step 2: Since the rank is $2$ , the eigenvalues are $1, 1,$ and $0$ .

Step 3: The sum of the eigenvalues is the trace of the matrix.

Step 4: Sum $= 1 + 1 + 0 = 2$ .

Answer: 2

Question 7 · Linear Algebra MCQ

Consider the following statements:

Assertion (A): The quadratic form $Q (x) = x^{T} B x$ with $B = (1021)$ is positive definite.

Reason (R): The eigenvalues of $B$ are both positive.

Which of the following is correct?

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: Definiteness of a quadratic form is determined by the eigenvalues of its symmetric part, not the original non-symmetric matrix.

Step 1: Check Reason (R): The eigenvalues of $B = (1021)$ are the roots of $(1 - λ)^{2} = 0$ , which are $1$ and $1$ . Both are positive. So R is true.

Step 2: Check Assertion (A): The symmetric part of $B$ is $A = \frac{B + B ^{T}}{2} = (1111)$ .

Step 3: The eigenvalues of $A$ are $0$ and $2$ . Since one eigenvalue is $0$ , the form is positive semi-definite, not positive definite. So A is false.

Answer: A is false but R is true.

Question 8 · Linear Algebra MCQ

Match the following matrix properties with their correct values for a

3 \times 3

centering matrix

M = I_{3} - \frac{1}{3} 1 1^{T}

.
List I: 1. Trace of

M

2. Determinant of

M

3. Eigenvalues of

M

List II: P.

0

and

1

(with multiplicities

2

and

1

) Q.

0

2

A.
1-P, 2-Q, 3-R
B.
1-R, 2-Q, 3-P
C.
1-Q, 2-R, 3-P
D.
1-R, 2-P, 3-Q

Correct Answer:

Step-by-Step Solution

Key idea: The centering matrix $M = I_{n} - \frac{1}{n} 1 1^{T}$ has specific trace, determinant, and eigenvalue properties.

Step 1: For $n = 3$ , the eigenvalues of $M$ are $1$ (with multiplicity $n - 1 = 2$ ) and $0$ (with multiplicity $1$ ). So eigenvalues are $0$ and $1$ . Matches P.

Step 2: The trace is the sum of eigenvalues: $1 + 1 + 0 = 2$ . Matches R.

Step 3: The determinant is the product of eigenvalues: $1 \times 1 \times 0 = 0$ . Matches Q.

Step 4: Correct matching is 1-R, 2-Q, 3-P.

Answer: 1-R, 2-Q, 3-P

Question 9 · Linear Algebra MCQ

Let $A = (a 2 2 a)$ . If $a \in {1, 2, 3, 4, 5}$ , for how many values of $a$ is the expression $x^{T} A x$ positive definite for all $x \neq = 0$ ?

A.
1
B.
2
C.
3
D.
4

Correct Answer:

Step-by-Step Solution

Key idea: A symmetric matrix is positive definite if and only if all its leading principal minors are strictly positive.

Step 1: The first leading principal minor is $a$ . For positive definiteness, we need $a > 0$ .

Step 2: The second leading principal minor is the determinant, $det (A) = a^{2} - 4$ . We need $a^{2} - 4 > 0$ , which implies $a > 2$ or $a < - 2$ .

Step 3: Combining $a > 0$ and $a > 2$ , we get $a > 2$ .

Step 4: Given the set $a \in {1, 2, 3, 4, 5}$ , the values satisfying $a > 2$ are $3, 4,$ and $5$ .

Step 5: There are exactly 3 such values.

Answer: 3

Question 10 · Linear Algebra MCQ

Consider the following statements:

Assertion (A): The expression $Q (x) = x^{T} B x$ with $B = (2052)$ is positive definite.

Reason (R): The diagonal elements of $B$ are positive.

Which of the following is correct?

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: The definiteness of a quadratic form is determined solely by the eigenvalues of its symmetric part, not the original non-symmetric matrix.

Step 1: Check Reason (R): The diagonal elements of $B$ are $2$ and $2$ , which are positive. So R is true.

Step 2: Check Assertion (A): We must symmetrize $B$ to find the true matrix governing the quadratic form. The symmetric part is $A = \frac{B + B ^{T}}{2} = (2 2.5 2.5 2)$ .

Step 3: The eigenvalues of $A$ are found from $(2 - λ)^{2} - 2. 5^{2} = 0 ⟹ 2 - λ = \pm 2.5 ⟹ λ = 4.5, - 0.5$ .

Step 4: Since one eigenvalue is negative, the expression is indefinite, not positive definite. So A is false.

Answer: A is false but R is true.

orthogonal projection and special matrices notes

Chapter Roadmap: Orthogonal, Projection and Special Matrices

Chapter Roadmap

Your Learning Journey

1. Projection Matrices and Quadratic Forms

2. Orthogonal and Involutory Matrices

By the end of this chapter, you will:

The Geometry of Quadratic Forms

The Geometry of Quadratic Forms

Geometric Intuition

The Symmetric Matrix Representation

The Symmetric Matrix Representation

Why does this work?

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

Orthogonal, Projection and Special Matrices Notes for GATE DA

Chapter Roadmap: Orthogonal, Projection and Special Matrices

Chapter Roadmap

Your Learning Journey

1. Projection Matrices and Quadratic Forms

2. Orthogonal and Involutory Matrices

By the end of this chapter, you will:

The Geometry of Quadratic Forms

The Geometry of Quadratic Forms

Geometric Intuition

The Symmetric Matrix Representation

The Symmetric Matrix Representation

Why does this work?

Classifying Quadratic Forms

Classifying Quadratic Forms

Orthogonal, Projection and Special Matrices: 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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