Sonia Ayers

2022-07-02

I don't claim at all to be an expert on this topic. In many (advanced) linear algebra textbooks for undergraduates, I usually find something about the "Jordan Canonical Form" of a matrix.

What is the purpose of such a form? I have taken a usual first-course in linear algebra (did another semester with Axler, but I don't claim to be an expert) and have taken abstract algebra (most familiar with group and ring theory) and have briefly skimmed through linear algebra books covering this material, but I don't quite understand the "big picture" idea, i.e., why is this useful in application? One person once told me it is the "most straightforward and useful algorithm for solving systems of linear equations, once you get beyond 3 variables or so," but maybe I'm missing something, since I usually don't see anything like what this person described to me in the linear algebra books I have. Most textbooks I've seen tend to have a more theoretical focus on this topic.

Also, any suggested texts which have good coverage on this topic would be very helpful.

What is the purpose of such a form? I have taken a usual first-course in linear algebra (did another semester with Axler, but I don't claim to be an expert) and have taken abstract algebra (most familiar with group and ring theory) and have briefly skimmed through linear algebra books covering this material, but I don't quite understand the "big picture" idea, i.e., why is this useful in application? One person once told me it is the "most straightforward and useful algorithm for solving systems of linear equations, once you get beyond 3 variables or so," but maybe I'm missing something, since I usually don't see anything like what this person described to me in the linear algebra books I have. Most textbooks I've seen tend to have a more theoretical focus on this topic.

Also, any suggested texts which have good coverage on this topic would be very helpful.

verzaadtwr

Beginner2022-07-03Added 17 answers

Two square matrices $A$ and $B$ are said to be similar, or conjugate, if there exists an invertible square matrix $P$ such that $A={P}^{-1}BP$. This is equivalent to saying that $A$ and $B$ represent the same linear transformation in different bases, with $P$ providing the change-of-basis matrix that relates them.

If one wants to solve a linear equation but is working in an inconvenient basis, it may help to change the basis to a more convenient one. Sometimes one can find a convenient basis by inspection, but in general one often changes the basis to obtain the Jordan canonical form of the desired matrix. For solving linear equations the Jordan canonical form is ideal, since (1) it has a very simple structure (upper triangular, and only 1-s just above the diagonal) and (2) it can be computed for any square matrix.

It is important for theoretical reasons to know that one can always find the Jordan canonical form of a square matrix. It simplifies many abstract proofs to assume a matrix in the proof is in Jordan canonical form. If you know a little abstract algebra, the Jordan canonical form is also of interest in the sense that it completely classifies the conjugacy classes of matrices over the complex numbers (and some other fields as well), and is a special case of a more general phenomenon regarding module homomorphisms.

However, as for more real-world purposes the Jordan canonical form is not ideal. The primary example of a real-world application would be solving a system of linear equations (for example, one that comes up when trying to solve a system of linear ODEs), and unfortunately the Jordan canonical form is not well-suited to this task in practice. The reason is that the Jordan canonical form is very sensitive to perturbations in the original matrix; that is, if an entry ${a}_{ij}$ in the matrix $A$ is perturbed to ${a}_{ij}+\u03f5$, it is very possible for the Jordan canonical form of the new matrix to be wildly different from the original Jordan canonical form. (That is, the Jordan canonical form is not numerically stable.)

The numerical instability of the Jordan canonical form makes it bad in real-life applications, where systems of linear equations arise from real-world data that always has a level of uncertainty. For this reason, in real-world applications one must abandon the Jordan canonical form for numerically stable algorithms. One example of such an algorithm is the Schur factorization, which also transforms (using unitary matrices) a matrix into a conjugate upper triangular matrix, and thus simplifies the solution of linear systems.

If one wants to solve a linear equation but is working in an inconvenient basis, it may help to change the basis to a more convenient one. Sometimes one can find a convenient basis by inspection, but in general one often changes the basis to obtain the Jordan canonical form of the desired matrix. For solving linear equations the Jordan canonical form is ideal, since (1) it has a very simple structure (upper triangular, and only 1-s just above the diagonal) and (2) it can be computed for any square matrix.

It is important for theoretical reasons to know that one can always find the Jordan canonical form of a square matrix. It simplifies many abstract proofs to assume a matrix in the proof is in Jordan canonical form. If you know a little abstract algebra, the Jordan canonical form is also of interest in the sense that it completely classifies the conjugacy classes of matrices over the complex numbers (and some other fields as well), and is a special case of a more general phenomenon regarding module homomorphisms.

However, as for more real-world purposes the Jordan canonical form is not ideal. The primary example of a real-world application would be solving a system of linear equations (for example, one that comes up when trying to solve a system of linear ODEs), and unfortunately the Jordan canonical form is not well-suited to this task in practice. The reason is that the Jordan canonical form is very sensitive to perturbations in the original matrix; that is, if an entry ${a}_{ij}$ in the matrix $A$ is perturbed to ${a}_{ij}+\u03f5$, it is very possible for the Jordan canonical form of the new matrix to be wildly different from the original Jordan canonical form. (That is, the Jordan canonical form is not numerically stable.)

The numerical instability of the Jordan canonical form makes it bad in real-life applications, where systems of linear equations arise from real-world data that always has a level of uncertainty. For this reason, in real-world applications one must abandon the Jordan canonical form for numerically stable algorithms. One example of such an algorithm is the Schur factorization, which also transforms (using unitary matrices) a matrix into a conjugate upper triangular matrix, and thus simplifies the solution of linear systems.

Cierra Castillo

Beginner2022-07-04Added 6 answers

Among other things the Jordan form can show that any Markov process must terminate and also helps to find the limiting state. Instructors usually motivate such processes by asking how to find equilibrium states in chemistry, but this also has applications to google's page rank system. Basically rank pages due to the largest entry in the limiting state.

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