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Quiz-Fall2010-1_2
Course: EML 4507, Fall 2010School: University of Florida
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4507 EML Fall 2010 Quiz 1-2 If you want, write your name in the back. Solution Problem 3: Matrix A given by [A]=8442 and the eigen values of [A] are: 0 and 10. Determine if it is: (a) positive-definite; or (b) Positive semi definite; or (c) Negative definite. Solution Since the eigne values are greater than or equal to zero (0), matrix A is positive semi definite. Illustration The quadratic form f=xy8442xy can...
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4507 EML Fall 2010
Quiz 1-2
If you want, write your name in the back.
Solution
Problem 3: Matrix A given by [A]=8442 and the
eigen values of [A] are: 0 and 10.
Determine if it is: (a) positive-definite; or (b) Positive
semi definite; or (c) Negative definite.
Solution
Since the eigne values are greater than or equal to
zero (0), matrix A is positive semi definite.
Illustration
The quadratic form f=xy8442xy can be expanded
as f=22x+y2. Then one can see f=0 for all 2x+y=0
or y=-2x. Thus, f=0 for combinations other than
x=y=0. However, f >0 is always satisfied, because it
is a perfect square.
I1=7,
I2=5+3+3=11,
I3=5
Hence, the cubic equation for principal stresses
is:
3-I12+I2-I3=0
or
3-72+11-5=0
-5-1-1=0
Problem 2: is It determined that n=5 is one of the
principal stresses for the state of stress in
Problem 1.
(i) Write the three linear equations you will use to
determine the corresponding principal direction.
Problem 1: Derive the (cubic) characteristic
equation for determining the principal
stresses of the stress matrix shown below.
Do NOT solve the equation.
(ii) Solve the above equations to determine the
principal direction. If there are multiple principal
directions for n=5, then show them in a parametric
form.
Solution
(3-5)-20-2(3-5)000(1-5)nxnynz=000
The first two equations are the same:
-2nx-2ny=0nx=-ny
The last equation yields nz=0
Solution
Calculate the invariants:
Then the principal direction is given by
121,-1,0
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