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

Orthogonal, Projection and Special Matrices short notes for GATE DA: 2 study cards covering concepts, formulas, shortcuts and exam traps, plus solved practice questions.

Projection & Quadratic Forms Cheat Sheet

1. Quadratic Forms

Symmetrize First: $A_{sy m} = \frac{B + B ^{T}}{2}$
Definiteness: Dictated by signs of eigenvalues of $A_{sy m}$ .

2. Projection Matrices

General: $P^{2} = P$ (Idempotent).
Orthogonal: $P^{2} = P$ AND $P^{T} = P$ .
Eigenvalues: Exactly $0$ and $1$ . Rank = number of $1$ s.

3. The Centering Matrix

Formula: $M = I_{n} - \frac{1}{n} 1 1^{T}$
Eigenvalues: $1$ (mult $n - 1$ ) and $0$ (mult $1$ ).
Action: Subtracts the mean from components.

4. Constrained Extrema

Max of $x^{T} A x$ subject to $x^{T} x = 1$ is $λ_{m a x} (A)$ .
Min of $x^{T} A x$ subject to $x^{T} x = 1$ is $λ_{m i n} (A)$ .

Special Matrices Cheat Sheet

1. Orthogonal Matrices ( $Q$ )

Definition: $Q^{T} Q = Q Q^{T} = I ⟹ Q^{- 1} = Q^{T}$ .
Determinant: $det (Q) = \pm 1$ .
Eigenvalues: Magnitude is exactly $1$ ( $∣ λ ∣ = 1$ ). Can be complex.
Geometry: Preserves vector lengths and angles.

2. Involutory Matrices ( $A$ )

Definition: $A^{2} = I ⟹ A = A^{- 1}$ .
Determinant: $det (A) = \pm 1$ .
Eigenvalues: Strictly $+ 1$ or $- 1$ .
Trace: Sum of eigenvalues (number of $+ 1$ s minus number of $- 1$ s).

3. High Powers Strategy

Involutory: $A^{2 k} = I$ , $A^{2 k + 1} = A$ .
Orthogonal/Periodic: Find smallest $p$ such that $A^{p} = I$ . Then $A^{n} = A^{n (mod p)}$ .

4. $2 \times 2$ Involutory Shortcut

If $A^{2} = I$ and $A \neq = \pm I$ , then $Tr (A) = 0$ and $det (A) = - 1$ .

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

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

Orthogonal, Projection and Special Matrices short notes for GATE DA: 2 study cards covering concepts, formulas, shortcuts and exam traps, plus solved practice

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

Projection & Quadratic Forms Cheat Sheet

Projection & Quadratic Forms Cheat Sheet

1. Quadratic Forms

2. Projection Matrices

3. The Centering Matrix

4. Constrained Extrema

Special Matrices Cheat Sheet

Special Matrices Cheat Sheet

1. Orthogonal Matrices (Q)

2. Involutory Matrices (A)

3. High Powers Strategy

4. 2×2 Involutory Shortcut

Orthogonal, Projection and Special Matrices: Solved Questions with Step-by-Step Explanations (2 Problems)

Step-by-Step Solution

Step-by-Step Solution

More short notes in this unit

A question from this chapter

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1. Orthogonal Matrices ( $Q$ )

2. Involutory Matrices ( $A$ )

4. $2 \times 2$ Involutory Shortcut