# Linear algebra Assignment Paper Out

#### Linear algebra

Linear algebra

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A1    S>0, 5×7-62 = -1,

So, not positive definite

A2 -1<0, so not positive definite.

A3 1>1, 1×100-10×10 = 0,

So, not positive definite

A4 1>, 1×101-10×10 = 1

So, it is positive definite

If x1 = 1, x2=-1, then this product is 0.

Question 18

Solutions:

K=ATA is symmetric positive definite if and only if A has independent columns

For, columns of A are independent. So ATA will be positive definite.

For, columns of A are independent. So ATA will be positive definite.

For, columns of A are independent. So ATA 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).

Question 14

1. Positive
2. Negative definite
3. Indefinite
4. Negative definite

Question 15

1. False
2. False
3. True
4. True

Question 32

Question 41

On the one hand, Ax =λMx is the same as CTACy =λy (writing M = RTR for C = R−1, and putting Rx = y). Then yTBy/yTy has its minimum value at λ1(B=CTAC), the least eigenvalue for the generalized eigenvector problem. On the other hand, this quotient is equal to xTAx/xTMx, which sometimes equals a 11/m11, e.g., when x equals the standard unit vector e1.

Problem Set 6.3

Question 2

Question 5

As ?=0 corresponds with ,   does not enter the picture

References

Bretscher, O. (2004). Linear Algebra with Applications, (3rd 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, (4th ed.). New York, NY: Prentice Hall.

Leon, S. J. (2006). Linear Algebra with Applications, (7th 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.

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