Questions tagged [matrices]

For any topic related to matrices. This includes: systems of linear equations, eigenvalues and eigenvectors (diagonalization, triangularization), determinant, trace, characteristic polynomial, adjugate and adjoint, transpose, Jordan normal form, matrix algorithms (e.g. LU, Gauss elimination, SVD, QR), invariant factors, quadratic forms, etc. For questions specifically concerning matrix equations, use the (matrix-equations) tag.

A matrix is a rectangular array of elements, usually numbers or variables, arranged in rows and columns. A matrix with $m$ rows and $n$ columns has $m \times n$ elements and is called an $m$ by $n$ matrix. Matrices are a part of .

Matrices can be added and subtracted. Furthermore, if they have compatible shapes, they can be multiplied. More precisely, given two matrices $A$ and $B$, the matrix $AB$ is defined when the number of columns of $A$ is equal to the number of rows of $B$. In particular, given a natural number $n$, any two matrices $A$ and $B$ with $n$ columns and $n$ rows can be multiplied in both ways (that is, both $AB$ and $BA$ exist).


For questions specifically concerning matrix equations, use the tag.

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How to show that $\det(AB) =\det(A) \det(B)$?

Given two square matrices $A$ and $B$, how do you show that $$\det(AB) = \det(A)\det(B)$$ where $\det(\cdot)$ is the determinant of the matrix?
Learner
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How do I tell if matrices are similar?

I have two $2\times 2$ matrices, $A$ and $B$, with the same determinant. I want to know if they are similar or not. I solved this by using a matrix called $S$: $$\left(\begin{array}{cc} a& b\\ c& d \end{array}\right)$$ and its inverse in terms…
user4681
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What are some applications of elementary linear algebra outside of math?

I'm TAing linear algebra next quarter, and it strikes me that I only know one example of an application I can present to my students. I'm looking for applications of elementary linear algebra outside of mathematics that I might talk about in…
user98602
67
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2 answers

Determinant of a rank $1$ update of a scalar matrix, or characteristic polynomial of a rank $1$ matrix

This question aims to create an "abstract duplicate" of numerous questions that ask about determinants of specific matrices (I may have missed a few): Characteristic polynomial of a matrix of $1$'s Eigenvalues of the rank one matrix…
67
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What is the largest eigenvalue of the following matrix?

Find the largest eigenvalue of the following matrix $$\begin{bmatrix} 1 & 4 & 16\\ 4 & 16 & 1\\ 16 & 1 & 4 \end{bmatrix}$$ This matrix is symmetric and, thus, the eigenvalues are real. I solved for the possible eigenvalues and,…
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Why is the determinant the volume of a parallelepiped in any dimensions?

For $n = 2$, I can visualize that the determinant $n \times n$ matrix is the area of the parallelograms by actually calculating the area by coordinates. But how can one easily realize that it is true for any dimensions?
ahala
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What is the difference between the Frobenius norm and the 2-norm of a matrix?

Given a matrix, is the Frobenius norm of that matrix always equal to the 2-norm of it, or are there certain matrices where these two norm methods would produce different results? If they are identical, then I suppose the only difference between them…
Ricket
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What are some mathematically interesting computations involving matrices?

I am helping designing a course module that teaches basic python programming to applied math undergraduates. As a result, I'm looking for examples of mathematically interesting computations involving matrices. Preferably these examples would be…
65
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The relation between trace and determinant of a matrix

Let $M$ be a symmetric $n \times n$ matrix. Is there any equality or inequality that relates the trace and determinant of $M$?
TPArrow
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Is the rank of a matrix the same of its transpose? If yes, how can I prove it?

I am auditing a Linear Algebra class, and today we were taught about the rank of a matrix. The definition was given from the row point of view: "The rank of a matrix A is the number of non-zero rows in the reduced row-echelon form of A". The…
Vivi
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Are all eigenvectors, of any matrix, always orthogonal?

I have a very simple question that can be stated without proof. Are all eigenvectors, of any matrix, always orthogonal? I am trying to understand Principal components and it is cruucial for me to see the basis of eigenvectors.
Bober02
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Matrices commute if and only if they share a common basis of eigenvectors?

I've come across a paper that mentions the fact that matrices commute if and only if they share a common basis of eigenvectors. Where can I find a proof of this statement?
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How to prove and interpret $\operatorname{rank}(AB) \leq \operatorname{min}(\operatorname{rank}(A), \operatorname{rank}(B))$?

Let $A$ and $B$ be two matrices which can be multiplied. Then $$\operatorname{rank}(AB) \leq \operatorname{min}(\operatorname{rank}(A), \operatorname{rank}(B)).$$ I proved $\operatorname{rank}(AB) \leq \operatorname{rank}(B)$ by interpreting $AB$…
user365
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Similar matrices and field extensions

Given a field $F$ and a subfield $K$ of $F$. Let $A$, $B$ be $n\times n$ matrices such that all the entries of $A$ and $B$ are in $K$. Is it true that if $A$ is similar to $B$ in $F^{n\times n}$ then they are similar in $K^{n\times n}$? Any help…
61
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Distance/Similarity between two matrices

I'm in the process of writing an application which identifies the closest matrix from a set of square matrices $M$ to a given square matrix $A$. The closest can be defined as the most similar. I think finding the distance between two given matrices…
Synex
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