(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

I am studying about eigenvalues and norms. I was wondering whether the way I understand them is correct.

2. Relevant equations

3. The attempt at a solution

The eigenvalue of a matrix those that satisfy [tex]Ax = \lambda x[/tex], where A is a matrix, x is an eigenvector, [tex]\lambda[/tex] is a scalar. The significant here is that from the perspective of the eigenvector, a matrix multiplication with A is the same as a scalar multiplication with lambda. When A is square, non-signular, and of rank n, A has n eigenvalues and n eigenvectors. The n eigenvectors forms a basis.

Suppose v_{i}is an eigenvector with associated eigenvalue [tex]\lambda_i[/tex]. Suppose an arbitrary vector y can be presented as:

[tex]y = a v_1 + bv_2[/tex]

Then

[tex]Ay = a\lambda_1v_1 + b\lambda_2v_2[/tex]

The magnitude (2-norm) of vector y in terms of the eigenvector basis is [tex]\sqrt{a^2+b^2}[/tex]. The magnitude of Ay is [tex]\sqrt{(a\lambda_1)^2 + (b\lambda_2)^2}[/tex]. The gain in magnitude is the ration between the two, which could be a mess to compute. If we check the gain in magnitude over all possible y, we get the 2-norm of matrix A.

Since the 2-norm could be messy to compute. Say we look at the [tex]\infty[/tex]-norm. With the infinity norm, the magnitude of y is the maximum value between a and b, and the magnitude of Ay is the maximum value between [tex]a\lambda_1[/tex] and [tex]b\lambda_2[/tex]. The gain in magnitude is again the ratio. For each y, the gain would one of the four possiblilities: [tex]\lambda_1[/tex], [tex]\lambda_2[/tex], [tex]\lambda_1a/b[/tex], [tex]\lambda_2b/a[/tex]. In the last two cases, the fraction a/b or b/a must be less than 1, because, it a > b in the first case, then the norm of y would be a, so its gain would have been [tex]\lambda_1[/tex] instead. Therefore, when all possible vector y are considered, the gain of matrix A must be the maximum between [tex]\lambda_1[/tex] and [tex]\lambda_2[/tex].

Topic 2: the eigenvalues of sum of matrices:

For a matrix A, there is a Jordan norm form which is an upper triangular matrix with the eigenvalues of A in its diagonal. A and its Jordan form J by an invertible matrix P in this fashion: [tex]AP = PJ[/tex]. Since det(AP)=det(A)det(P), det(A) = [tex]\prod \lambda_i[/tex].

Suppose we have matrix A and B. A has Jordan form J such that AP = PJ. B has Jordan form K such that BQ = QK. Then the determinant of A+B is:

det(A + B) = det( PJP^{-1}+ QKQ^{-1})

If P happens to equal Q, then:

det(A + B) = det( P( J+K )P^{-1}) = [tex]\prod (\lambda_j + \lambda_k)[/tex]

One situation where P can equal Q is when A is the identity matrix. So the matrix M := I+B would have eigenvalues [tex]\lambda_m = 1+\lambda_b[/tex].

Is this explanation correct?

- Thanks

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# Eigenvalues matrix help

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