chapter 7

Question 1

symmetric

Answer 1

A(transpose)=A

Question 2

orthogonal matrix

Answer 2

if U(transpose)U=I

i.e. U(inverse)=U(transpose)

Question 3

orthogonally diagonalizable

Answer 3

if there exists an orthogonal nxn matrix U such that U(inverse)AU=U(transpose)AU is a diagonal matrix

Question 4

the spectral theorem

Answer 4

let A be an nxn real symmetric matrix. Then:

• the eigenvalues of A are real
• eigenvectors corresponding to distinct eigenvalues are orthogonal
• A is orthogonally diagonalizable

Question 5

function Q is a ____________

Answer 5

quadratic form

Question 6

positive definite if…..

Answer 6

Q(x)>0 for all x not= 0

Question 7

negative definite if…..

Answer 7

Q(x)<0 for all x not= 0

Question 8

indefinite

Answer 8

if Q(x) takes on both positive and negative values

Question 9

if the columns of A are lin dependent then _________

Answer 9

0 is an eigenvalue