Matrices Questions - Symmetric, Minimal Polynomial, Bilinear Form, Jordan

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In summary, the conversation is discussing different concepts related to matrices and their properties. The first question is asking if any symmetric matrix can be diagonalized with an orthogonal change of basis. The second question is about the minimal polynomial of a real matrix and whether it splits into distinct linear factors. The third question is asking if a real inner product is an example of a bilinear form. The final question is asking if two complex matrices that are similar can have different Jordan normal forms. The conversation also includes some discussion and clarification about the fourth question.
  • #1
Ted123
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Is any symmetric matrix diagonalisable with an orthogonal change of basis?

Does the minimal polynomial of any real matrix split into distinct linear factors?

Is a real inner product an example of a bilinear form?

Could 2 complex matrices which are similar have different Jordan normal forms?
 
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  • #2
So, what did you try already??
 
  • #3
micromass said:
So, what did you try already??

I just want to know 'yes' or 'no' out of interest
 
  • #4
For the first one, let A be symmetric. Let u be an eigenvector with eigenvalue [itex]\lambda[/itex] and let w be an eigenvector with eigenvalue [itex]\mu[/itex]. Can you calculate

[tex]<Au,w>[/tex]

in several ways?
 
  • #5
I think YES, NO, YES, NO respectively. Is that correct?
 
  • #6
Ted123 said:
I think YES, NO, YES, NO respectively. Is that correct?

Why do you think that?
 
  • #7
I think the 4th question is worded slightly ambiguously; it's probably safe to assume that it means "different up to reordering of the Jordan blocks" (in which case you're correct). But if it doesn't, then [tex]\begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 1 \\ 0 & 0 & 1\end{pmatrix}[/tex] and [tex]\begin{pmatrix} 1 & 1 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{pmatrix}[/tex] would be an example of similar matrices with different JNFs.

I know the first 3 are right - yes, no, yes but for the 4th one, do you think that the answer to the last one is no?
 
  • #8
Ted123 said:
I think the 4th question is worded slightly ambiguously; it's probably safe to assume that it means "different up to reordering of the Jordan blocks" (in which case you're correct). But if it doesn't, then [tex]\begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 1 \\ 0 & 0 & 1\end{pmatrix}[/tex] and [tex]\begin{pmatrix} 1 & 1 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{pmatrix}[/tex] would be an example of similar matrices with different JNFs.

I know the first 3 are right - yes, no, yes but for the 4th one, do you think that the answer to the last one is no?

If you mean "different up to reordering of the Jordan blocks" then you are correct. Similar matrices have the same Jordan canonical form then.

This can be seen by putting [itex]A=S^{-1}BS[/itex]. If C is the Jordan basis, then

[tex]CAC^{-1}[/tex]

is the Jordan canonical form of A. Now, can you find easily the Jordan canonical form of B (just plug the expressions into each other).
 

Related to Matrices Questions - Symmetric, Minimal Polynomial, Bilinear Form, Jordan

1. What is a symmetric matrix?

A symmetric matrix is a square matrix where the elements are symmetric about the main diagonal. This means that the element at row i and column j is equal to the element at row j and column i. In other words, a symmetric matrix A satisfies A[i,j] = A[j,i].

2. How do you find the minimal polynomial of a matrix?

The minimal polynomial of a matrix is the smallest degree monic polynomial that the matrix satisfies. To find the minimal polynomial, we can use the Cayley-Hamilton theorem, which states that every square matrix satisfies its own characteristic polynomial. Thus, the minimal polynomial can be found by setting the characteristic polynomial equal to 0 and solving for the variable.

3. What is a bilinear form?

A bilinear form is a function that takes two vectors as inputs and produces a scalar as output. It is linear in both of its inputs, meaning that it follows the properties of linearity, such as scaling and addition. An example of a bilinear form is the dot product of two vectors.

4. What is the Jordan form of a matrix?

The Jordan form of a matrix is a specific form that a matrix can be transformed into by a similarity transformation. It is a diagonal matrix with blocks of Jordan blocks along the diagonal. A Jordan block is a square matrix with a constant value along the main diagonal and 1s along the superdiagonal. The Jordan form is useful for understanding the properties of a matrix and for solving systems of linear equations.

5. How are matrices used in data analysis and machine learning?

Matrices are used extensively in data analysis and machine learning for tasks such as data transformation, dimensionality reduction, and modeling. They are especially useful for handling large datasets and performing operations such as matrix multiplication and inversion, which are important for many algorithms. Matrices also allow for efficient computation and storage of data, making them a crucial tool in data analysis and machine learning.

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