Linear Algebra proof, diagonalization

In summary: But is that matrix B?I am sorry for the confusion. I was just trying to say that the matrix given is A and I am trying to use the formula for a similar matrix written in terms of A. I was just trying to use the formula to find B or N in the problem.In summary, the problem states that N is a 2x2 matrix such that N^2 = 0. The task is to prove that either N = 0 or N is similar to the matrix ((0,0), (1,0)). This can be done by finding a basis for R^2 and showing that N is similar to a matrix with this basis. The basis can be found by setting N(v1
  • #36
Suppose [itex]N=\begin{pmatrix}1 & 1 \\ -1 & -1 \end{pmatrix}[/itex].
Does it satisfy the problem criteria?
Are V_1 and V_2 as you're suggesting?
 
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  • #37
Do they satisfy the transformation or the ((1,1),(-1,-1)) * V_1= V_2...no for the second one if V_1 = (0,1)...completely lost now.
 
  • #38
Suppose V_1=(1,1).
Does it fit?

What would V_2 be?
And what would N^2 be?
 
  • #39
Are you just picking arbitrary V_1? shouldn't it be the first column?
 
  • #40
No, it's not arbitrary. :)

Why should it be the first column?
 
  • #41
We had always used columns in class and from the book...and the basis are all columns and everything we have been writing or I at least have been writing is in columns.

Which is why I would've thought (1,-1) is V1 and why I asked if it was arbitrary...if I was using rows i'd expect (1,1).
 
  • #42
That's why I adjusted post #36, to avoid confusion.

So what is:
[tex]N\cdot \begin{pmatrix}1 \\ 1\end{pmatrix}=\begin{pmatrix}1 & 1 \\ -1 & -1 \end{pmatrix} \cdot \begin{pmatrix}1 \\ 1\end{pmatrix}[/tex]

And what do you get if you multiply N with the resulting vector?

What are therefore V_1 and V_2?
 
  • #43
(a_11+a_12,a_21+a_22) as a 2x1 matrix if we do not know what v_1 and v_2 are..

V_1 and V_2 are both (1,1) or the 2x2 matrix of ((1,1),(1,1))
 
  • #44
Did you calculate [itex]\begin{pmatrix}1 & 1 \\ -1 & -1 \end{pmatrix} \cdot \begin{pmatrix}1 \\ 1\end{pmatrix}[/itex]?
 
  • #45
yes (2,-2) column vector...also tried to solve the problem using the matrix ((a,b),(c,d)) and got a=+-d
 
  • #46
hedgie said:
yes (2,-2) column vector

Yes. And N*(2,-2)?


hedgie said:
...also tried to solve the problem using the matrix ((a,b),(c,d)) and got a=+-d

a=-d would be right, but how did you get a=d?
 
  • #47
I don't, I only got a=-d, sorry typo.

((1,-1),(1,-1))*(2,-2)=(0,0) and (V_1 V_2) = ((2,-2),(2,-2))
 
  • #48
hedgie said:
I don't, I only got a=-d, sorry typo.

Yes.


hedgie said:
((1,-1),(1,-1))*(2,-2)=(0,0) and (V_1 V_2) = ((2,-2),(2,-2))

No: (V_1 V_2) = ((1,1),(2,-2))
 
  • #49
Why would it be the first row and the column that i am multiplying it against transposed to a row?
 
<h2>1. What is diagonalization in linear algebra?</h2><p>Diagonalization is a process in linear algebra where a square matrix is transformed into a diagonal matrix by finding a set of eigenvectors and eigenvalues.</p><h2>2. Why is diagonalization important in linear algebra?</h2><p>Diagonalization is important because it simplifies calculations involving the matrix, making it easier to solve systems of equations and perform other operations.</p><h2>3. How do you prove diagonalizability of a matrix?</h2><p>To prove diagonalizability of a matrix, you need to show that the matrix has a full set of linearly independent eigenvectors. This can be done by finding the eigenvalues and corresponding eigenvectors and showing that they form a basis for the matrix.</p><h2>4. Can every matrix be diagonalized?</h2><p>No, not every matrix can be diagonalized. A matrix can only be diagonalized if it has a full set of linearly independent eigenvectors. If there are not enough eigenvectors, the matrix cannot be diagonalized.</p><h2>5. What are the applications of diagonalization in real life?</h2><p>Diagonalization has many applications in fields such as physics, engineering, and economics. It is used to solve systems of differential equations, model population growth, and analyze financial data, among other things.</p>

1. What is diagonalization in linear algebra?

Diagonalization is a process in linear algebra where a square matrix is transformed into a diagonal matrix by finding a set of eigenvectors and eigenvalues.

2. Why is diagonalization important in linear algebra?

Diagonalization is important because it simplifies calculations involving the matrix, making it easier to solve systems of equations and perform other operations.

3. How do you prove diagonalizability of a matrix?

To prove diagonalizability of a matrix, you need to show that the matrix has a full set of linearly independent eigenvectors. This can be done by finding the eigenvalues and corresponding eigenvectors and showing that they form a basis for the matrix.

4. Can every matrix be diagonalized?

No, not every matrix can be diagonalized. A matrix can only be diagonalized if it has a full set of linearly independent eigenvectors. If there are not enough eigenvectors, the matrix cannot be diagonalized.

5. What are the applications of diagonalization in real life?

Diagonalization has many applications in fields such as physics, engineering, and economics. It is used to solve systems of differential equations, model population growth, and analyze financial data, among other things.

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