invertible matrices are positive definite?

If a matrix appears as $A=B^{-1}$, is it automatically positive definite?

Invertible matrices have full rank, and so, nonzero eigenvalues, which in turn implies nonzero determinant (as the product of eigenvalues).

How to Stay Positive When It Feels Like Your Life is Falling Apart


  1. apexis

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    How to Stay Positive When It Feels Like Your Life is Falling Apart

    1. Be Grateful

    Even if it seems everything is going wrong, there are things that are going right. Every day, write down all the things you are grateful for having happened to you today. Be descriptive of each item. Feel the joy those things bring you. Even if it’s something really small like your car started this morning. Be grateful for it. This will help you to stay focused on the positive.

    2. Look for the Silver Linings

    Sometimes events that initially look bad turn out to be really good things in the end. I can’t remember the saying but it was something like every black cloud has a silver lining. Here’s an example, maybe you need to look for a new job because your company is downsizing. If you look for the silver lining in that, you can see that it’s making you leave a job you didn’t like that much anyway. You can now find that job that you really want to do or pursue a lifelong dream or even just take some time off to enjoy life before going back to find another job. By looking for the silver lining then bad things can actually become really good things.

    3. Often the Greatest Doubts Occur Just Before a Breakthrough

    This is a quote by Lama Surya Das. If you can think of your difficulties as being necessary to make a breakthrough to the new life you want to lead, it will make it easier. You’re breaking free of your old life on the way to your new, more exciting life.

    4. Look at the Big Picture and Focus

    Decide on what direction you want your life to head. Think about what areas you want to explore, what experiences you want to feel, what do you want to accomplish in your life? What kind of person do you want to be? Then start taking steps towards those goals. Even if it’s just tiny steps every day, stay focused on where you want to go, and keep moving forwards. Then the little things you’re going through right now won’t matter as much because you have an exciting new direction to focus on instead.

    5. Discouragement is Just a Trick

    Here’s a quote I really like about not letting yourself get discouraged. “Discouragement is a negative emotion with more than one trick up its dark sleeve. It tricks you into mentally or emotionally dwelling in the very place you want to leave. Drop all such sorrow permanently by daring to see through this deception of the nconscious mind. You have a destination far beyond where you find yourself standing today” – Guy Finley Keep this in mind as you continue to take steps towards your goals. You do have a destination far beyond where you are today.

    6. Keep Reading Inspirational Articles and Maybe Join a Forum

    If you keep reading inspirational articles and books, it will motivate you to stay positive. You will see that everyone goes through rough patches but that they do make it through. It will be worth it in the end. Also, there are some good forums you can join where even just reading the other posts can motivate you to stay positive. Or, if you want to make a posting, you can gain support from others as well. Who knows, you may even be able to answer some one else’s post and help them out too.

    7. Take a Break Sometimes

    Forget about your problems and just go out and do something fun. Go to a movie, go out with some fun, positive thinking friends, go to a play, go for a bike ride, just do something. Get out and just enjoy yourself for a little bit. Sometimes, we just get so wrapped up in our problems that we forget to have fun. Taking a break will re-energize you and you may find that things aren’t as bad as you thought.

     



  2. diggersjohn33


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    Point #1 is a biggie for me.

    If you believe, you can achieve.

     


  3. clickfund


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    This is a great post, so true that you can make things boulders on the beach or pebbles in the sand

     


  4. Amanda@Mindsettable


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    I really like your points, and I think when things really seem tough, #3 is especially good to keep in mind.

     

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number theory – For any positive integer $k$, there exists a prime $p$ such that ${x choose k}equiv -1pmod{p}$ has an integer solution.

Here is a conjecture of mine. Prove that for any positive integer $k$, there exists a prime $p$ such that $${x choose k}equiv -1pmod{p}$$ has an integer solution.

When $k$ is odd, we can take $x$ to be $p-1$, the question can be easily solved because $${p-1 choose k}equiv (-1)^kequiv -1pmod{p}.$$ However, when $k$ is even, I cannot find any pattern. Here are results for $k=4$ and $k=6$.

$$x=30implies {x choose 4}equiv -1pmod{193},$$
$$x=34implies {x choose 6}equiv -1pmod{97}.$$

It Just Feels Good to Be Positive

Your attitude can really make a difference in the outcome of various situations in your life. The way you approach the things that go on in your daily walk can really impact you in ways you might not have thought about. Imagine you are going along your normal routine and you encounter a challenge along the way. How you handle that particular challenge can really mean the difference between a success and a failure.
Being positive in the time of a challenge can really help make your life easier. You can experience less stress because your mindset is set on positivity and not negativity. There’s just something about being negative that drains all of your energy right out of you and puts you in a horrible frame of mind and mood which is not healthy. Being positive releases less stress and puts your mindset in one of a “can do” attitude which results in better results and not being stressed out. It’s amazing how a simple smile or laugh can change your attitude. When I am faced with a challenge, sometimes I just break out into laughter. Sometimes you have to laugh to avoid crying or being upset. It has really helped, plus when you frown, you get more wrinkles (smile).

The next time you are in a stressful situation or life just happens, take time to make every effort to keep a positive mindset in your daily walk. It can make you more productive and reduce stress. Have a wonderful day!

 

modular arithmetic – Any easier way to get the quotient and remainder from a minus number divided by a positive one?

negative number divided by positive number, what would be remainder?

I’ve read these answers linked above, but I don’t feel I’m answered enough.
$$-27 = underbrace{-6}_qcdot underbrace{5}_d + underbrace{3}_r$$
They say this, but I don’t think I can use it later when I need to divide some large minus number with another large positive number, such as -777777 with 439. Positive number division is easy enough that no matter how young you are, even grammar school students could solve it with enough time given. I was wondering if there’s any that kind of easy method to get the quotient and remainder from the minus numbers divided by the positive numbers.

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Thank you for your time and let me know if you have any questions about them….I would answer as far as my knowledge about them goes.

Let me know in the comments if other people have tried them and what’s their opinion about them!

real analysis – Flux through the positive part of a sphere centered at $(0,0,1)$

Let $$B={(x,y,z):x^2+y^2+(z-1)^2<4, zgeq 0}$$ and consider the vector field $$F:(x,y,z)mapsto(x^3,y^3,z)$$

I want to compute the flux of $F$ through $partial B.$

We have $$text{Div}F=3x^2+3y^2+1$$ and so by the divergence theorem we could compute the flux as $$int_B 3x^2+3y^2+1 dx dy dz$$
but this triple integral does not look very friendly to me using spherical coordinates, because the sphere is centered in $(0,0,1).$

Another approach is by computing $$partial B ={x^2+y^2leq 3, z=0}cup{x^2+y^2+(z-1)^2=4, z>0}$$

I can parametrize ${x^2+y^2+(z-1)^2=4, z>0}$ as $$x=2sin phi cos psi$$ $$y=2 sin phi sin psi$$ $$z= 2 cos phi +1$$ $$phiin(0,pi), psi in (0,2 pi)$$ but it does not seem to make calculations easier.

For infinitely even positive integers k for which ϕ(n)=k has no positive integer solution

Show that for infinitely even positive integers k for which ϕ(n)=k has no positive integer solution n

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