Linear Algebra Questions
LINEAR ALGEBRA 2
Question 14
A 1 S>0, 5×7-6 2 = -1,
So, not positive definite
A 2 -1<0, so not positive definite.
A3 1>1, 1×100-10×10 = 0,
So, not positive definite
A 4 1>, 1×101-10×10 = 1
So, it is positive definite
If x 1 = 1, x 2 =-1, then this product is 0.
Question 18
Solutions:
K=A T A is symmetric positive definite if and only if A has independent columns
For, columns of A are independent. So A T A will be positive definite.
For, columns of A are independent. So A T A will be positive definite.
For, columns of A are independent. So A T A will not be positive definite.
Question 7
Since a matrix is positive-definite if and only if all its eigenvalues are positive, and since the
eigenvalues of A −1 are simply the inverses of the, eigenvalues of A, A −1 is also positive definite
(the inverse of a positive number is positive).
LINEAR ALGEBRA 3
Question 14
a. Positive
b. Negative definite
c. Indefinite
d. Negative definite
Question 15
a. False
b. False
c. True
d. True
LINEAR ALGEBRA 4
Question 32
Question 41
On the one hand, Ax =λMx is the same as C T ACy =λy (writing M = R T R for C = R −1 , and putting
Rx = y). Then y T By/y T y has its minimum value at λ 1 (B=CTAC), the least eigenvalue for the
generalized eigenvector problem. On the other hand, this quotient is equal to x T Ax/x T Mx, which
sometimes equals a 11/m11, e.g., when x equals the standard unit vector e1.
Problem Set 6.3
Question 2
LINEAR ALGEBRA 5
Question 5
As ��=0 corresponds with , does not enter the picture
LINEAR ALGEBRA 6
References
Bretscher, O. (2004). Linear Algebra with Applications, (3 rd ed.). New York, NY: Prentice Hall.
Farin, G., & Hansford, D. (2004). Practical Linear Algebra: A Geometry Toolbox. London: AK
Peters.
Friedberg, S. H., Insel, A. J., & Spence, L. E. (2002). Linear Algebra, (4 th ed.). New York, NY:
Prentice Hall.
Leon, S. J. (2006). Linear Algebra with Applications, (7 th ed.). New York, NY: Pearson Prentice
Hall.
McMahon, D. (2005). Linear Algebra Demystified. New York, NY: McGraw–Hill Professional.
Zhang, F. (2009). Linear Algebra: Challenging Problems for Students. Baltimore, MA: The
Johns Hopkins University Press.
