#### Linear algebra

Linear algebra

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SAMPLE ANSWER **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).

**Question 14**

- Positive
- Negative definite
- Indefinite
- Negative definite

**Question 15**

- False
- False
- True
- True

** ****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**

**Question 5**

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

**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.

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