Complex numbers + linear algebra = :S

In summary, the conversation is about finding complex numbers and vectors that satisfy the equation Av=zv, where A is a given matrix and v and z are indeterminate variables. This is equivalent to finding eigenvalues and eigenvectors of the matrix. By setting v as a column vector (x,y) and equating it to Av, a system of two equations can be formed. This leads to a quadratic equation, which has two roots that are complex numbers. To find these complex numbers, one can refer to examples of finding eigenvalues.
  • #1
philnow
83
0

Homework Statement



Find all complex numbers Z (if any) such that the matrix: (it is 2 by 2)

(2)(-1)
(4)(2)

multiplied by a vector V = ZV has a nonzero solution V.

part b)

for each Z that you found, find all vectors V such that (the same matrix)*V=ZV.

The Attempt at a Solution



I'm not sure that I even know what this question is asking... could anyone clarify this a little?
 
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  • #2
And please don't mistake my lack of effort for laziness, I genuinely don't understand the question.
 
  • #3
The question is "solve Av=zv".

(A is the matrix you were given)
(v and z are the indeterminate variables you're solving for)
(v is a vector variable)
(z is a complex variable)
 
  • #4
In other words, as Hurkyl clarified, it's asking you for eigenvalues and eigenvectors of your matrix. Search on those keywords if you need examples.
 
  • #5
So I'm looking for complex numbers such that, when multiplied by a vector, it gives the same results as the matrix multiplied by the same vector?
 
  • #6
philnow said:
So I'm looking for complex numbers such that, when multiplied by a vector, it gives the same results as the matrix multiplied by the same vector?

Yes, the number times the vector should be the same as the matrix times the vector.
 
  • #7
I googled eigenvectors/values but I don't see the relevance to this question :s

How would I find a set of complex numbers that multiplies the same way as a matrix of real numbers? Any hints please?
 
Last edited:
  • #8
I'm surprised you didn't see the relevance. But look, let v be column vector (x,y). So z*v is column vector (z*x,z*y) What is Av written out as a column vector? Now equate the two column vectors so you have a system of two equations, write them down. What condition does z have to satisfy to get a solution where x and y are not both zero? It's a quadratic. It has two roots. They are complex. See if you can find them. Look back at examples of finding eigenvalues again. Because that is what you are doing.
 
  • #9
That makes a lot of sense, thanks Dick :)
 

1. What are complex numbers and how are they used in linear algebra?

Complex numbers are numbers that include both a real and an imaginary component. They are typically represented in the form a + bi, where a is the real part and bi is the imaginary part. In linear algebra, complex numbers are used to represent vectors and matrices in two dimensions, allowing for more complex calculations and solutions.

2. How do you add and subtract complex numbers?

To add or subtract complex numbers, you simply combine the real and imaginary parts separately. For example, (3 + 2i) + (1 + 4i) would result in (3 + 1) + (2i + 4i) = 4 + 6i. When subtracting, you would follow the same process but with subtraction instead of addition.

3. Can complex numbers be multiplied and divided?

Yes, complex numbers can be multiplied and divided. When multiplying complex numbers, you use the FOIL method just like with binomials. For example, (3 + 2i)(1 + 4i) would result in 3 + 12i + 2i + 8i^2 = -5 + 14i. To divide complex numbers, you would use the conjugate of the denominator to simplify the expression.

4. How are complex numbers graphed in the complex plane?

Complex numbers are graphed on the complex plane, which is similar to the traditional x-y plane but with the real numbers represented on the x-axis and the imaginary numbers on the y-axis. The point (a,b) on the complex plane represents the complex number a + bi.

5. What is the connection between complex numbers and linear transformations?

Complex numbers can be used to represent linear transformations, which are functions that take in a vector and output a new vector. For example, a rotation transformation can be represented by a complex number with a magnitude of 1 and an angle of rotation as the argument. This allows for more complex transformations to be represented and calculated in linear algebra.

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