Lx=d: SundayResearch

All the following poems describe a number. 産医師異国に向こう．産後薬なく産に産婆四郎二郎死産．産婆さんに泣く．ご礼には早よ行くな． Yes, I have a number. How I want a drink, alcoholic of course, after the heavy lectures involving quantum mechanics. Sir, I send a rhyme excelling．In sacred truth and rigid spelling． Numerical spirits elucidate, For me, the lesson's dull weight. If Nature gain, Not you complain, Tho' Dr. Johnson fulminate. Que j'aime à faire apprendre un nombre utile aux sages! Immortel Archimède, artiste ingénieur, Qui de ton jugement peut priser la valeur? Pour moi, ton problème eut de pareils avantages. Wie, O dies π． Macht ernstlich so vielen viele Müh!  Lernt immerhin, Jünglinge, leichte Verselein.  Wie so zum Beispiel dies dürfte zu merken sein! The German version mentioned about the number --- π． These are all Mnemonic for π. Japanese uses sounds of numbers, but, other languages uses the number of words. From Yes(3), I(1) have(4) a(1) number(6), you can find 3.1416 (rounded). I fo

If eigenvectors of a matrix A are independent, it is a happy property. Because the matrix A can be diagonalized with a matrix S that column vectors are eigenvectors of A . For example, Why this is a happy property of A? Because I can find A's power easily. A^{10} is not a big deal. Because Λ is a diagonal matrix and power of a diagonal matrix is quite simple. A^{10} = SΛ^{10} S^{-1} Then, why if I want to compute power of A ? That is the same reason to find eigenvectors. Eigenvectors are a basis of a matrix. A matrix can be represented by a single scalar. I repeat this again. This is the happy point, a matrix becomes a scalar. What can be simpler than a scalar value. But, this is only possible when the matrix S's columns are independent. Because S^{-1} must be exist. Now I come back to my first question. Is the λ's multiplicity related with the number of eigenvectors? This time I found this has the name. Geometric multiplicity (GM): the number of in

I had a question regarding the relationship between multiplicity of eigenvalue and eigenvectors. I am more interested in eigenvalue's multiplicity than the value itself. Because if eigenvalue has multiplicity, the number of independent eigenvectors ``could'' decrease. My favorite property of eigen-analysis is that is a transformation to simpler  basis. Here, simpler  means a matrix became a scalar. I even have a problem to understand a 2x2 matrix, but a scalar has no problem, or there is no simpler thing than a scalar. A x = λ x  means the matrix A  equals λ, what a great simplification! My question is  If λ has multiplicity, are there still independent eigenvectors for the eigenvalue? My intuition said no. I can compute an eigenvector to a corresponding eigenvalue. But, I think I cannot compute the independent eigenvectors for one eigenvalue. For instance, assume 2x2 matrix that has λ = 1,1, how many eigenvectors? one? Recently I found this is related with di