Math (1, 2, 2)

Question 1

define Eigenvalue

Answer 1

A scalar λ ∈ R, or a factor by which the vector (which stays on its original span) gets stretched or squished

An eigenvector x is a vector that satisfies this equation for the eigenvalue.

Question 2

Define Spectrum

Question 3

define Eigenvector

Answer 3

0 ≠ x ∈ R^n. vectors that stay on their span after a linear transformation.

Question 4

eigen vector and value equation

Answer 4

an eigen value of A is a scalar

Question 5

What is an eigenspace W associated with an eigenvalue ?

Answer 5

The vector subspace of Rn defined by W = {x 2 Rn : (A - I)x = 0}.

Question 6

True or False: Eigenvectors associated with distinct eigenvalues of a symmetric matrix are orthogonal.

Answer 6

True.

Question 7

What is the characteristic polynomial of a symmetric matrix A of order n?

Answer 7

pA(t) = tn + n-1tn-1 + … + (-1)n det A.

Question 8

What is the degree of the characteristic polynomial for a symmetric matrix of order n?

Answer 8

n.

Question 9

What does it mean if an eigenvalue has multiplicity m()?

Answer 9

It is the number of times the eigenvalue  appears as a root of the characteristic polynomial.

Question 10

What is an eigenbasis?

Answer 10

An orthonormal basis of Rn formed by eigenvectors of a symmetric matrix.

Question 11

Fill in the blank: The dimension of the eigenspace W is equal to the _______ of the eigenvalue .

Answer 11

multiplicity m().

Question 12

What is the trace of a matrix A?

Answer 12

The sum of the diagonal entries, denoted as tr A.

Question 13

In the context of eigenvalues, what does the Rank-Nullity Theorem state?

Answer 13

dim W + dim Im T = n.

Question 14

What is the implication of a symmetric matrix A having n distinct eigenvalues?

Answer 14

It indicates that there are at most n independent eigenvectors.

Question 15

How can one find an eigenpair (; x)?

Answer 15

First find the eigenvalue  by solving det(A - I) = 0, then find the eigenvector x by solving (A - I)x = 0.

Question 16

What is the relationship between the discriminant of the characteristic polynomial and the roots?

Answer 16

The discriminant indicates that all roots of the polynomial are real.

Question 17

What does Proposition 1251 state about the dimension of eigenspaces?

Answer 17

dim W = m() for each eigenvalue .

Question 18

What are the vectors associated with the eigenvalue 2?

Answer 18

(0; 0; 1) and (p3; p3; 0)

These vectors form part of the eigenbasis for R3.

Question 19

What is the dimension of W1?

Answer 19

1

Question 20

What is the dimension of W3?

Answer 20

2

Question 21

What forms an eigenbasis of R3?

Answer 21

The vectors (p3; 3; 0), (0; 0; 1), and (1; r2; 0)

These vectors are crucial for understanding the structure of the space.

Question 22

What is the Gram-Schmidt orthonormalization process used for?

Answer 22

To turn a set of linearly independent vectors into an orthonormal basis.

Question 23

What is the formula for the first normalized vector x~1?

Answer 23

x~1 = x1 / ||x1||

Question 24

What is the auxiliary vector y2 defined as?

Answer 24

y2 = x2 - Pspan{ x~1 }(x2)

Question 25

What does it mean if a symmetric matrix is invertible?

Answer 25

Its inverse matrix is also symmetric.

Question 26

What condition must be satisfied for a symmetric matrix A to be invertible?

Answer 26

All its eigenvalues must be non-zero.

Question 27

What is the relationship between the eigenvalues of a matrix and its inverse?

Answer 27

The eigenvalues of the inverse matrix are the reciprocals of the eigenvalues of the original matrix.

Question 28

What does it mean for a square matrix B to be orthogonal?

Answer 28

BTB = I

Question 29

List the equivalent conditions for a square matrix B to be orthogonal.

Answer 29

* B is orthogonal * B has orthonormal rows * B has orthonormal columns * B is invertible, with B^{-1} = BT

Question 30

What does the lemma state about a square matrix B having orthonormal rows?

Answer 30

B has orthonormal rows if and only if BBT = I.

Question 31

What is the determinant of an orthogonal matrix?

Answer 31

The determinant is either 1 or -1.

Question 32

What is the spectral decomposition of a symmetric matrix A?

Answer 32

A = BΛBT, where B is orthogonal and Λ is a diagonal matrix of eigenvalues.

Question 33

How is the determinant of a symmetric matrix related to its eigenvalues?

Answer 33

det A = λ1 * λ2 * ... * λn, where λi are the eigenvalues.

Question 34

What is the significance of the orthogonal matrix B in the context of diagonalization?

Answer 34

It consists of the normalized eigenvectors associated with the eigenvalues.

Question 35

True or False: If a symmetric matrix has distinct eigenvalues, its eigenvectors can form an orthogonal basis.

Answer 35

True

Question 36

What happens if two vectors x1 and x2 are linearly independent?

Answer 36

They can be orthonormalized using the Gram-Schmidt process.

Question 37

How do you define the normalized eigenvector x_i?

Answer 37

x_i = normalized eigenvector associated with eigenvalue λ_i.

Question 38

What condition must hold for a symmetric matrix A to be orthogonally diagonalizable?

Answer 38

It must have n distinct eigenvalues.

Question 39

What is the condition for the inequality involving xi and xj when i ≠ j?

Answer 39

xi ? xj when i ≠ j ## Footnote This condition ensures that the vectors are distinct.

Question 40

What is the significance of the orthogonal matrix in diagonalization?

Answer 40

The orthogonal matrix can be the matrix whose columns are normalized eigenvectors associated to distinct eigenvalues.

Question 41

What is an eigenvalue?

Answer 41

An eigenvalue is a scalar associated with a linear transformation represented by a matrix.

Question 42

What does the determinant of a matrix represent?

Answer 42

The determinant represents the scaling factor of the transformation described by the matrix.

Question 43

What is a symmetric matrix?

Answer 43

A symmetric matrix is a square matrix that is equal to its transpose.

Question 44

What is a quadratic form?

Answer 44

A quadratic form is a polynomial of degree two in several variables.

Question 45

How is a quadratic form represented using a symmetric matrix?

Answer 45

f(x) = x^T A x, where A is a symmetric matrix.

Question 46

What is a monomial of degree m?

Answer 46

A monomial of degree m is a function of the form f(x1, ..., xn) = k(x1^p1 * x2^p2 * ... * xn^pn) where the sum of the degrees equals m.

Question 47

What defines a form?

Answer 47

A form is a sum of monomials of the same degree.

Question 48

What is the relationship between quadratic forms and symmetric matrices?

Answer 48

There is a bijective correspondence between quadratic forms and symmetric matrices.

Question 49

What is a linear form?

Answer 49

A linear form is the sum of monomials of first degree.

Question 50

True or False: All square matrices have real eigenvalues.

Answer 50

False ## Footnote Eigenvalues can be complex for non-symmetric matrices.

Question 51

Fill in the blank: A function f : R^n → R is a ______ if it is a sum of monomials of the same degree.

Answer 51

form

Question 52

What are the coefficients of the squares in a quadratic form?

Answer 52

The elements of the diagonal of the symmetric matrix A.

Question 53

What is the characteristic of a quadratic form?

Answer 53

It is the sum of monomials of second degree.

Question 54

What does Riesz's Theorem state about linear forms?

Answer 54

Linear forms are linear functions.

Question 55

What is the general form of a quadratic form expanded?

Answer 55

f(x) = a11x1^2 + a22x2^2 + ... + 2a12x1x2 + ...

Question 56

What is the symmetric matrix associated with the quadratic form f(x1, x2) = x2 + x2 - 4x1x2?

Answer 56

A = [[1, -2], [-2, 1]]

Question 57

What is the determinant of a matrix A when eigenvalues are λ1 and λ2?

Answer 57

det A = λ1 * λ2

Question 58

What is the formula for the determinant of a symmetric matrix with eigenvalues?

Answer 58

det A = Π λi for i = 1 to n

Question 59

What is the definition of a quadratic form f : Rn ! R?

Answer 59

f (x) = Pn i=1 ix²

Question 60

What is the classification of a quadratic form based on its sign?

Answer 60

* Positive semi-definite: f (x) ≥ 0 for all x ∈ Rn * Negative semi-definite: f (x) ≤ 0 for all x ∈ Rn * Positive definite: f (x) > 0 for all 0 ≠ x ∈ Rn * Negative definite: f (x) < 0 for all 0 ≠ x ∈ Rn * Indefinite: There exist x, x0 ∈ Rn such that f (x) < 0 < f (x0)

Question 61

True or False: A quadratic form f is negative definite if and only if -f is positive definite.

Answer 61

True

Question 62

What are the conditions for a symmetric matrix A regarding positive and negative semi-definiteness?

Answer 62

* Positive semi-definite: x'Ax ≥ 0 for all x ∈ Rn * Negative semi-definite: x'Ax ≤ 0 for all x ∈ Rn * Positive definite: x'Ax > 0 for all 0 ≠ x ∈ Rn * Negative definite: x'Ax < 0 for all 0 ≠ x ∈ Rn

Question 63

What does it mean for a positive definite matrix to be invertible?

Answer 63

A positive definite matrix is invertible and its inverse is also positive definite.

Question 64

What is a necessary condition for a positive semi-definite matrix to be invertible?

Answer 64

It must be positive definite.

Question 65

What is the relationship between the eigenvalues of a symmetric matrix and its definiteness?

Answer 65

* Positive definite: all eigenvalues are strictly positive * Positive semi-definite: all eigenvalues are non-negative

Question 66

What does the determinant of a positive definite matrix indicate?

Answer 66

det A > 0

Question 67

What does the determinant of a positive semi-definite matrix indicate?

Answer 67

det A ≥ 0

Question 68

What is Cholesky factorization?

Answer 68

A = LLT where L is a lower triangular matrix with strictly positive diagonal entries.

Question 69

Fill in the blank: A quadratic form f (x) = Pn i=1 ix² is positive semi-definite if and only if _______.

Answer 69

i ≥ 0 for every i

Question 70

Fill in the blank: A quadratic form f (x) = Pn i=1 ix² is positive definite if and only if _______.

Answer 70

i > 0 for every i

Question 71

What is the Sylvester-Jacobi criterion used for?

Answer 71

To determine the definiteness of a symmetric matrix using principal minors.

Question 72

What are principal minors in the context of a symmetric matrix?

Answer 72

The determinants of principal submatrices obtained by eliminating rows and columns.

Question 73

What is the significance of leading principal minors?

Answer 73

They are the determinants of leading principal submatrices obtained by eliminating the last k rows and columns.

Question 74

Example: What is the determinant of the matrix A = [[1, 3, 2], [10, 1, 2], [4, 5, 7]]?

Answer 74

det A = -101

Question 75

What condition must be satisfied for the diagonal entries aii of a positive definite matrix?

Answer 75

aii > 0 for all i = 1, ..., n

Question 76

What condition must be satisfied for the diagonal entries aii of a positive semi-definite matrix?

Answer 76

aii ≥ 0 for all i = 1, ..., n

Question 77

Fill in the blank: A positive semi-definite matrix A is such that x'Ax = 0 if and only if _______.

Answer 77

Ax = 0

Question 78

What is the determinant of the matrix A if det A1 = 1, det A2 = -29, and det A3 = -101?

Answer 78

det A = -101 ## Footnote The leading principal minors are: det A1 = 1, det A2 = -29, det A3 = -101.

Question 79

What terminology was first used in a decomposition of a quadratic form in a sum of squares by Francesco Brioschi?

Answer 79

Leading principal minors ## Footnote Brioschi established this terminology in 1856.

Question 80

What does Theorem 1279 (Brioschi) state about a symmetric matrix A with non-zero leading principal minors?

Answer 80

There exists an upper triangular matrix C with unit diagonal entries such that x' Ax = Σ det Ak z^2 ## Footnote Where z = Cx.

Question 81

What form does the matrix C take in Brioschi's Theorem for n = 2?

Answer 81

C = [[1, c], [0, 1]] ## Footnote Where c ∈ R.

Question 82

What are the conditions for a symmetric matrix A to be positive definite according to Proposition 1280?

Answer 82

All its principal minors are strictly positive ## Footnote This is a necessary condition for positive definiteness.

Question 83

What does Proposition 1281 (Sylvester-Jacobi criterion) state about a symmetric matrix A?

Answer 83

A is positive definite if all leading principal minors are strictly positive; negative definite if they change sign starting with a negative sign; indefinite if their signs do not respect these conditions ## Footnote This criterion simplifies the checking of matrix definiteness.

Question 84

How can one conclude that a symmetric matrix is not definite or semi-definite according to Proposition 1280?

Answer 84

Exhibit any principal minor that violates the positivity conditions ## Footnote This approach is a preliminary analysis based on previous propositions.

Question 85

In Example 1282, what is the associated symmetric matrix of the quadratic form f(x1, x2, x3)?

Answer 85

A = [[1, 2, 3], [1, 1, 2], [0, 3, 2]] ## Footnote The matrix represents the quadratic form f(x1, x2, x3) = x1^2 + 2x2^2 + x3^2 + (x1 + x3)x2.

Question 86

What are the leading principal minors of matrix A in Example 1282?

Answer 86

det A1 = 1 > 0, det A2 > 0, det A3 = 2 > 0 ## Footnote This implies that the quadratic form is positive definite.

Question 87

What is the criterion for a symmetric matrix to be positive semi-definite according to Proposition 1283?

Answer 87

All its principal minors, leading or not, are positive ## Footnote This criterion is more computationally intensive than for positive definiteness.

Question 88

What is the condition under which the matrix A = [[0, 0], [0, a22]] is positive semi-definite?

Answer 88

a22 ≥ 0 ## Footnote Although the leading principal minors can be positive, the overall condition requires a22 to be non-negative.