Orthogonal, Projection and Special Matrices Notes for GATE DA: Concepts, Formulas, Worked Examples & Practice

Orthogonal, Projection and Special Matrices notes for GATE DA: 20 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 (2 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

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

More notes in this unit

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Orthogonal, Projection and Special Matrices Notes for GATE DA: Concepts, Formulas, Worked Examples & Practice

Orthogonal, Projection and Special Matrices notes for GATE DA: 20 study cards covering concepts, formulas, shortcuts and exam traps, plus solved practice ques

A question from this chapter

Question 1

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

Question 2

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

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