I'm trying to understand the difference between Upper Triangular and Diagonal Matrices for square matrices. There is a theorem that says every matrix can be made into an upper triangular matrix. Now if my matrix as the same number of eigenvectors as its dimension then I can make a Diagonal matrix by changing the basis to the eigenspace. So in this case, why will I ever prefer to make a matrix Upper Triangular(and not diagonal) when I can make it diagonal?
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$\begingroup$ Why would you change the matrix in the first place? What's your intention? Do you want to solve a linear system of equations or is it the characteristic matrix of a linear dynamical system? As it stands, this question is a good candidate for closure becasue of lack of context $\endgroup$– polfosolCommented Jul 6, 2019 at 16:22
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1$\begingroup$ Sometimes we are interested in bases which are orthonormal. Even if a matrix is diagonalisable in other bases, it may not be diagonalisable in an orthonormal basis. $\endgroup$– user1551Commented Jul 6, 2019 at 16:24
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Perhaps that you are missing the fact that every diagonal matrix is upper triangular too. So, if a matrix is diagonalizable, it is, by definition, similar to a diagonal matrix. Otherwise, it is not similar to a diagonal matrix, but it is still similar to an upper triangular one (at least over an algebraically closed field, such as $\mathbb C$).
answered Jul 6, 2019 at 16:23
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$\begingroup$ I meant why will I ever prefer to make a matrix Upper Triangular(and not diagonal) when I can make it purely diagonal? $\endgroup$ Commented Jul 6, 2019 at 17:08
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$\begingroup$ For no reason that I can imagine, other than the fact that sometimes a matrix is similar to a triangular one, but not to a diagonal one. $\endgroup$ Commented Jul 6, 2019 at 17:12