Chapter 5: Orthogonality

Question 1

Theorem 5.3.1 Properties of inner products and norms in Rn. Let u, v, w ∈ Rn and c ∈ R. Then

Answer 1

(a) u · v = v · u (commutativity of inner product)
(b) u · (v + w) = u · v + u · w (distributivity of inner product)
(c) (cu) · v = c(u · v)
(d) ||v||2 = v · v
(e) kvk ≥ 0 and kvk = 0 if and only if v = 0
(f) kcvk = |c|kvk

Question 2

Theorem 5.3.5 Let S = {v1, v2, . . . , vk} be an orthogonal set of non-zero vectors in Rn. Then

Answer 2

S is a linearly independent set.

Question 3

Coefficients for orthogonal basis in
v = c1v1 + c2v2 + · · · + ckvk,

Answer 3

ci =(v · vi)/(vi· vi)
, for i = 1, . . . , k

Question 4

Orthogonal matrix

Answer 4

columns for an orthonormal set

Question 5

Theorem 8.2.1 A square matrix Q is orthogonal if and only if

Answer 5

Q^−1 = QT

Question 6

Theorem 10.4.3 Let Q be an n × n matrix. The following are equivalent,

Answer 6

(a) Q is orthogonal
(b) ||Qx| = ||x|| for all x ∈ Rn
(c) Qx · Qy = x · y for all x, y ∈ Rn

Question 7

Theorem Ex 8.2.3 Let Q be an n × n orthogonal matrix. Then,

Answer 7

(a) Q^−1 is orthogonal
(b) det Q = ±1
(c) If λ is an eigenvalue of Q, then |λ| = 1
(d) If Q1 and Q2 are orthogonal n × n matrices then so is Q1Q2

Question 8

Definition Let A be an n × n matrix. A is orthogonally diagonalisable if

Answer 8

there exists an orthogonal matrix Q and a diagonal
matrix D such that Q^TAQ = D.

Question 9

Theorem 8.2.4 and 5.5.7 If A is a real symmetric matrix then

Answer 9

a) The eigenvalues of A are all real.
b) Eigenvectors corresponding to distinct eigenvalues are orthogonal.

Question 10

Theorem 8.2.2 Principal Axes Theorem If following conditions are equivalent for an n × n matrix A.

Answer 10

a) A has an orthonormal set of n eigenvectors
b) A is orthogonally diagonalisable
c) A is symmetric

Question 11

Definition A quadratic form in n variables is a function f : Rn →
R of the form

Answer 11

f(x) = x
TAx = a11x^2_1 + a22x_2^2 + · · · +annx^2_n + ∑i<j 2aijxixj
where A is a symmetric n × n matrix and x∈ Rn. A is the matrix
associated with f .

Question 12

x^TAx = 1 represents an ellipse if

Answer 12

λ1 > 0 and λ2 > 0

Question 13

x^TAx = 1 has no graph if

Answer 13

λ1 < 0 and λ2 < 0

Question 14

x^TAx = 1 represents an hyperbola if

Answer 14

λ1 and λ2 have opposite signs.

Question 15

Positive definite square matrix

Answer 15

symmetric and all its eigenvalues λ are positive (negative), that is λ > 0

Question 16

Negative definite square matrix

Answer 16

Symmetric and all its eigenvalues λ are negative, that is λ < 0

Question 17

Semidefinite

Answer 17

If the eigenvalues λ ≥ 0 (λ ≤ 0) then the matrix positive (negative) semidefinite

Question 18

Indefinite

Answer 18

eigenvalues take on both positive
and negative values

Question 19

Positive definite iff

Answer 19

xTAx > 0 for every x =/= 0 in Rn
(signature is n)

Question 20

Positive semidefinite iff

Answer 20

xTAx ≥ 0 for every x =/= 0 in Rn
(signature=rank)

Question 21

Negative definite iff

Answer 21

x
TAx < 0 for every x 6= 0
in Rn
(signature is −n)

Question 22

Rank

Answer 22

p + n where p is the number of positive eigenvalues and n is the number of negative eigenvalues

Question 23

Signature

Answer 23

p-n where p is the number of positive eigenvalues and n is the number of negative eigenvalues

Question 24

Negative semidefinite iff

Answer 24

xTAx ≤ 0 for every x =/= 0 in Rn
(signature= −rank)

Question 25

Indefinite iff

Answer 25

xTAx takes on both positive and negative values. (−rank

Question 26

Let u and v be vectors in C^n. Then u · v =

Answer 26

conjugate, transpose of u * v.

Question 27

Theorem 8.7.3 Let A, B denote complex n × n matrices and c ∈ C Then

Answer 27

a) (A^H)^H = A b) (A + B)^H = A^H + B^H c) (cA)^H = c conjugate A^H d) (AB)^H = B^H A^H

Question 28

Hermitian matrix

Answer 28

A square complex matrix A is called hermitian (selfadjoint) if A^H = A.

Question 29

Theorem 8.7.5 Let A be a complex n × n hermitian matrix.

Answer 29

a) Every eigenvalue of a hermitian matrix A is a real number. b) The eigenvectors corresponding to distinct eigenvalues of A are orthogonal.

Question 30

Unitary matrix

Answer 30

A square complex matrix U is called unitary if U^−1 = U^H.

Question 31

Theorem 8.7.6 Let U ∈ Mn×n(C) (the space of complex n × n matrices). The following are equivalent:

Answer 31

a) U is unitary matrix b) The columns of U form an orthonormal basis for Cn with respect to the complex dot product c) ||Ux|| = ||x|| for every x ∈ Cn d) Ux · Uy = x · y for every x, y ∈ Cn

Question 32

Unitary diagonalisable

Answer 32

A square complex matrix A is called unitary diagonalisable if there exists a unitary matrix U and diagonal matrix D such that UH AU = D.

Question 33

Spectral Theorem

Answer 33

Every hermitian matrix A is unitarily diagonalisable.

Question 34

Are all unitarily diagonalisable matrices Hermitian?

Answer 34

No

Question 35

Are all hermitian matrices unitarily diagonalisable?

Answer 35

Yes

Question 36

Principal axis theorem asserts that

Answer 36

a real matrix is symmetric if and only if it is orthogonally diagonalisable

Question 37

Are all orthogonally diagonalisable matrices symmetric?

Answer 37

Yes

Question 38

Are all symmetric matrices orthogonally diagonalisable?

Answer 38

Yes

Question 39

Normal matrix

Answer 39

A square complex matrix A is called normal if A^H A = A A^H

Question 40

Theorem 8.7.9 A square complex matrix A is unitarily diagonalisable if and only if

Answer 40

A is normal, that is A^H A = A A^H

Question 41

Matrices that are normal

Answer 41

Every hermitian matrix, every unitary matrix, and every skew hermitian matrix (A^H = −A) is normal.