This website is no longer maintained by Yu. ST is the new administrator. Linear Algebra Problems by Topics The list of linear algebra problems is available here. Subscribe to Blog via Email Enter your email address to subscribe to this blog and receive notifications of new posts by email.
Sponsored Links. Search for:. MathJax Mathematical equations are created by MathJax. Step by Step Explanation. More in Linear Algebra Quiz 1. Math Spring The best answers are voted up and rise to the top. Stack Overflow for Teams — Collaborate and share knowledge with a private group. Create a free Team What is Teams? Learn more. Why is it important for a matrix to be square? Ask Question. Asked 3 years, 5 months ago. Active 3 years, 5 months ago.
Viewed 34k times. Rodrigo de Azevedo Beneschan Beneschan 1, 2 2 gold badges 7 7 silver badges 10 10 bronze badges. Matrices with complex values are just as useful and of course a real matrix can be considered as a special case of a complex matrix. Also as N8tron said there is a very powerful concept that applies to non-square matrices, namely the "singular value decomposition" SVD - though to understand what it is and what it's good for, you need to learn about the concepts you mentioned for square matrices first, and the SVD may not be covered at all in a "first" linear analysis course.
Add a comment. Active Oldest Votes. So really your question amounts to: Why are maps from a space to itself - as opposed to maps from a space to something else - particularly interesting? Noah Schweber Noah Schweber k 18 18 gold badges silver badges bronze badges. I probably would have enjoyed linear algebra a lot more if it had been accompanied with these sorts of explanations. C Monsour C Monsour 7, 1 1 gold badge 7 7 silver badges 30 30 bronze badges.
Ethan Bolker Ethan Bolker Or a set of vectors with its image: by how much was the volume of this parallelepiped stretched? Arnaud Mortier Arnaud Mortier The domain and codomain have to be equal for you to compose the transformation with itself.
While a square matrix might sometimes represent a map between distinct spaces of the same dimension, everything that the question asks about are intrinsic properties of maps from a space to itself, so it doesn't feel particularly relevant to me to bring up a different semantic case where determinants, characteristic polynomials, and eigenvectors don't naturally apply.
We note that, as this is a diagonal matrix, this might be a useful form for the matrix to be in. Interestingly, the matrix is no longer diagonal. At this point, it is possible to recognize a pattern. For the odd powers, this is not the case, since there is a nonzero entry in the lower-left corner and the bottom-right entry becomes negative. There are many related topics that bolster the justification for studying matrix exponentiation. When working with a square matrix, it is clear that repeatedly multiplying such a matrix by itself will generally lead to results that are successively more complicated to calculate given the large numbers involved, as we have seen in several of the examples above.
It is therefore advantageous to be able to reduce the complexity of these calculations as much as possible. Under certain circumstances, it is possible to diagonalize a matrix, which significantly reduces the complexity of calculating its integer powers.
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Please contact your portal admin. Lesson Explainer: Power of a Matrix Mathematics. Conventional algebra is commutative over multiplication.
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