# 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.

35,940 questions
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### What's an intuitive way to think about the determinant?

In my linear algebra class, we just talked about determinants. So far I’ve been understanding the material okay, but now I’m very confused. I get that when the determinant is zero, the matrix doesn’t ...
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### If $AB = I$ then $BA = I$

If $A$ and $B$ are square matrices such that $AB = I$, where $I$ is the identity matrix, show that $BA = I$. I do not understand anything more than the following. Elementary row operations. Linear ...
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### A matrix and its transpose have the same set of eigenvalues

Let $\sigma(A)$ be the set of all eigenvalues of $A$. Show that $\sigma(A) = \sigma(A^T)$ where $A^T$ is the transposed matrix of $A$.
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### If the field of a vector space weren't characteristic zero, then what would change in the theory?

In the book of Linear Algebra by Werner Greub, whenever we choose a field for our vector spaces, we always choose an arbitrary field $F$ of characteristic zero, but to understand the importance of the ...
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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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### 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 ...
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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?
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### Polynomial equations $p(A, B) = 0$ for matrices that ensure $AB = BA$

Let $k$ be a field with characteristic different from $2$, and $A$ and $B$ be $2 \times 2$ matrices with entries in $k$. Then we can prove, with a bit art, that $A^2 - 2AB + B^2 = O$ implies $AB = BA$,...
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### Why is that if every row of a matrix sums to 1, then the rows of the inverse matrix sums to 1 too?

Why is that if every row of a matrix sums to $1$ then the rows of its inverse matrix sum to $1$ too? For example, consider $$A=\begin{pmatrix} 1/3 & 2/3 \\ 3/4 & 1/4 \end{pmatrix}$$ then ...
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### Are the eigenvalues of $AB$ equal to the eigenvalues of $BA$? (Citation needed!)

First of all, am I being crazy in thinking that if $\lambda$ is an eigenvalue of $AB$, where $A$ and $B$ are both $N \times N$ matrices (not necessarily invertible), then $\lambda$ is also an ...
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### Intuitive explanation of a positive-semidefinite matrix

What is an intuitive explanation of a positive-semidefinite matrix? Or a simple example which gives more intuition for it rather than the bare definition. Say $x$ is some vector in space and $M$ is ...
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 ...
Let $M$ be a symmetric $n \times n$ matrix. Is there any equality or inequality that relates the trace and determinant of $M$?