Math Is Fun Forum

What does 'SVD' stand for?

You are welcome to create a thread in Help Me if you need any more help -- we'll do our best to respond to any troubles you might have.

Ah, so \prime is the right one to use... and to think all this time I've been deliberately trying to avoid apostrophes in LaTeX!

This is really just an application of diagonalisation. Let's consider the following system:

Suppose that

and that

Then, if ' denotes differentation with respect to t, we write x' = Ax. We now make a change of variable. Put x = Py, where P is invertible, and:

and

Then, differentiating with respect to t, x' = Py' so that the system x' = Ax reduces to Py' = APy. That is:

Py' = APy  =>  y' = P⁻¹APy.

Does P⁻¹AP look familiar to you?

Not a problem. Sometimes the best way to become unstuck on a maths problem is to simply do something else.

Yes, all complex numbers can be represented in the form a + bi, where a, b are real numbers. With them, we can define addition and multiplication as follows:

Some examples:

We also have some other properties, such as conjugation:

There exist even more interesting algebraic structures, such as the quaternions (which are not commutative) and the octonions (which are not associative!).

Yes, that's correct. In fact, you already completed the 'proof' under your 'solving' heading -- just do everything backwards.

Use the Euclidean algorithm.

That integral is not expressible in terms of elementary functions. That integral evaluates to:

i.e. the imaginary error function, which you can read about here. There are, however, alternative ways of expressing that integral in terms of an infinite series.

It would be best to consult a table of Laplace Transforms.

Your first can be split using partial fractions, and then using the linearity of the Laplace transform.

For the second, what is w? If w is a constant, then you can read this result off directly from the link above.

If you're not permitted to use a table, then you can compute the inverses directly using the formula:

where gamma = Re(s), constructed so that gamma is always greater than every singularity of F(s) (this requires some knowledge of complex analysis). However, the nature of your problem seems to indicate that they want you to use a table of Laplace transforms.

I can't do anything more for you without additional information about a,b. Is this what you were looking for?

Your problem is simply dealing with the brackets (and it seems you made a small error in the formula).

i.e.