Linear algebra Assignment Paper Out

Linear algebra

Linear algebra

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SAMPLE ANSWER            Question 14

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=A^TA is symmetric positive definite if and only if A has independent columns

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

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

For, columns of A are independent. So A^TA 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 C^TACy =λy (writing M = R^TR for C = R⁻¹, and putting Rx = y). Then y^TBy/y^Ty 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 x^TAx/x^TMx, 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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