0
votes
0answers
23 views

Ortonormal basis of unitary operator and its spectral decomposition - check my solution.

Dear fellow mathematicians, I'm trying to do a linear algebra exercise, but I have no idea whether I have a correct plan of solution. Here is the problem: Find orthonormal eigenbasis (not sure if ...
0
votes
1answer
37 views

Eigen vectors of the matrix whose columns are eigen vectors of the original matrix

Consider a matrix $A$ of dimension $n$X$n$ whose eigen vectors are $y_1,y_2,y_3,...,y_n$ and are linearly independent. What are the properties of the eigen vectors of the matrix $P$ whose columns are ...
2
votes
0answers
52 views

Calculating the signature of a matrix

The task is the following: Consider $\mathbb{R}^2$ equipped with the canonical dot-product $\langle \cdot , \cdot \rangle$, and also the symmetrical bilinear form $$\beta(u,v) := \left\langle u,\ ...
1
vote
2answers
23 views

orthonormal vector properties

I have noticed a matrix property that is outlined below: I have a set of n orthonormal eigenvectors that form a basis in Rn. If these vectors are combined to form an nxn matrix where each column is ...
1
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1answer
122 views

Eigenvalues of a $3\times3$ orthogonal matrix

Can anyone give me an example of 3x3 orthogonal matrix with complex eigenvalue.
1
vote
4answers
59 views

How to find vector that is orthonormal on other.

I have to generate a Q matrix for Schur decomposition and I have the first column, let's it is the following: \begin{bmatrix}1/√3\\1/√3\\1/√3 \end{bmatrix} Now I need to find the second column that ...
2
votes
4answers
728 views

PCA - Image compression

I have 2 questions related to principal component analysis: The first is, how do you prove that the principal components matrix forms a orthonormal basis? Are the eigenvalues always orthogonal? The ...
0
votes
1answer
73 views

Normal matrices with orthogonal basis

we have a theorem that says that each REAL normal matrix can be written in terms of an orthonormal basis, so that it has its eigenvalues down the diagonal and 2x2 matrices of the form $\begin{pmatrix} ...
2
votes
1answer
1k views

What's the trick for proving one eigenvalue of orthogonal matrix is $-1$ if the determinant is $-1$?

Obviously, the magnitude of the orthogonal matrix is 1, which is easy to prove.. However, I wonder how can one prove that the eigenvalue of an orthogonal matrix is $-1$, if the determinant of this ...
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vote
1answer
187 views

Verifying Orthogonality of Eigenvectors

How do you 'verify' the orthogonality of the eigenvectors of a matrix, let's say ${\bf A}$ , for example? I came across the result that a matrix ${\bf A}$ has orthogonal eigenvectors if ${\bf ...
3
votes
1answer
117 views

Orthogonality, Maximization and Eigen-Solution

I Have read that for a matrix of reals $Y$ and a p.s.d matrix $B$ that the Maximum of $ f(Y)=Tr(Y^TBY)$ subject to $Y^TY = I$ is achieved when $span(Y)$ equals the span of the first $d$ ...
0
votes
1answer
300 views

Orthonormalization of non-hermitian matrix eigenvectors

I have been told that orthonormalization of the eigenvectors of a non-hermitian matrix has to use a different definition of inner product than when the matrix is hermitian. Why is this so, and how do ...
4
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2answers
2k views

Orthonormal Eigenbasis

I am a little apprehensive to ask this question because I have a feeling it's a "duh" question but I guess that's the beauty of sites like this (anonymity): I need to find an orthonormal eigenbasis ...