Tagged Questions

For any topic related to matrices. This includes: systems of linear equations, eigenvalues and eigenvectors (diagonalization, triangularization), determinant, trace, characteristic polynomial, adjugate, transpose, Jordan normal form, matrix algorithms (e.g. LU, Gauss elimination, SVD, QR), ...

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for any ring $A$ the matrix ring $M_n(A)$ is simple if and only if $A$ is simple

Let integer $n\geq 1$. I have obtained that for any field $k$, the matrix ring $M_n(k)$ is simple, i.e., $M_n(k)$ contains no nonzero proper two sided ideals. Now I want to prove that: for any ring $A$...
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Eigenvalues for the rank one matrix $uv^T$

Suppose $A=uv^T$ where $u$ and $v$ are non-zero column vectors in ${\mathbb R}^n$, $n\geq 3$. $\lambda=0$ is an eigenvalue of $A$ since $A$ is not of full rank. $\lambda=v^Tu$ is also an eigenvalue of ...
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Elegant proofs that similar matrices have the same characteristic polynomial?

It's a simple exercise to show that two similar matrices has the same eigenvalues and eigenvectors (my favorite way is noting that they represent the same linear transformation in different bases). ...
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what are pivot numbers in LU decomposition? please explain me in an example

studying many pages like wikipedia, wolfram, Mathworks, Math Stack Exchange, lu-pivot, LU_Decomposition I couldn't find exactly whart are the pivot numbers in a specific example: for example what are ...
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if eigenvalues are positive, is the matrix positive definite?

If the matrix is positive definite, then all its eigenvalues are strictly positive. Is the converse also true? That is, if the eigenvalues are strictly positive, then matrix is positive definite? Can ...
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Can commuting matrices $X,Y$ always be written as polynomials of some matrix $A$?

Consider square matrices over a field $K$. I don't think additional assumptions about $K$ like algebraically closed or characteristic $0$ are pertinent, but feel free to make them for comfort. For any ...
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Determinant of transpose?

$$\det(A^T) = \det(A)$$ Using the geometric definition of the determinant as the area spanned by the columns could someone give a geometric interpretation of the property? Thanks!
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Eigenvalues and power of a matrix

Let $A$ be an n×n matrix with eigenvalues $\lambda_i, i=1,2,\dots,n$. Then $\lambda_1^k,\dots,\lambda_n^k$ are eigenvalues of $A^k$. I was wondering if $\lambda_1^k,\dots,\lambda_n^k$ are all the ...
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How to calculate gradient of $x^TAx$

I am watching the following video lecture: https://www.youtube.com/watch?v=G_p4QJrjdOw In there, he talks about calculating gradient of $x^{T}Ax$ and he does that using the concept of exterior ...
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How to prove $\det(e^A) = e^{\operatorname{tr}(A)}$?

Prove $$\det(e^A) = e^{\operatorname{tr}(A)}$$ for all matrices $A \in \mathbb{C}_{n×n}$.
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if $A=AB-BA$ then $A^n=0$?

Let matrix $A_{n\times n}$, be such that there exists a matrix $B$ for which $$AB-BA=A$$ Prove or disprove there exsit $n\in N^{+}$such $$A^n=0,$$ I know $$tr(A)=tr(AB)-tr(BA)=0$$ then I can't....
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how many unique patterns exist for a NxN grid

I'm trying to figure out if there is a way to determine how many unique patterns exist for a given NxN grid if you choose N points on the grid. For example, for a 2x2 grid we can get two unique ...
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Why does a diagonalization of a matrix B with the basis of a commuting matrix A give a block diagonal matrix?

I am trying to understand a proof concerning commuting matrices and simultaneous diagonalization of these. It seems to be a well known result that when you take the eigenvectors of A as a basis and ...
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Characterization of the trace function

We know that the trace of a matrix is a linear map for all square matrices and that $\operatorname{tr}(AB)=\operatorname{tr}(BA)$ when the multiplication makes sense. On the Wikipedia page for trace,...
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What are the eigenvalues of matrix that have all elements equal 1? [duplicate]

As in subject: given a matrix $A$ of size $n$ with all elements equal exactly 1. What are the eigenvalues of that matrix ?
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Determining eigenvalues, eigenvectors of $A\in \mathbb{R}^{n\times n}(n\geq 2)$.

Let $a$ and $b$ be distinct nonzero real numbers and let $A\in \mathbb{R}^{n\times n}(n\geq 2)$ with each diagonal entry equal to $a$ and each off-diagonal entry equal to $b$. Determine all ...
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Rank of skew-symmetric matrix

Is there a succinct proof for the fact that the rank of a non-zero skew-symmetric matrix ($A = -A^T$) is at least 2? I can think of a proof by contradiction: Assume rank is 1. Then you express all ...
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Is sum of two orthogonal matrices singular?

I am trying to solve following problem. Let $A,B\in \mathbb{R}^{n\times n}$ be an orthogonal matrices and $\det(A)=-\det(B)$. How can be proven that $A+B$ is singular? I could start with ...
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Prove $\det(kA)=k^n\det A$

Let $A$ be a $n \times n$ invertible matrix, prove $\det(kA)=k^n\det A$. I really don't know where to start. Can someone give me a hint for this proof?
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Quaternions vs Axis angle

Whats the use of representing rotation with quaternions compared to normal axis angle representation? I've been trying to learn quaternions and they make enough sense but as far as I can tell ...
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$\left \| \cdot \right \|$ is an induced norm. If $\left \| A \right \|<1$, how to show that $I-A$ is nonsingular and …?

The induced norm $\left \| \cdot \right \|$ is defined for a matrix $A\in\mathbb{C}^{n\times n}$ as $\left \| A \right \|=\sup_{||x||=1} \left \| Ax \right \|$. If $\left \| A \right \|<1$, show ...
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Constructing a graph from a degree sequence

Let's say I'm given several degree sequences like {4,3,3,2,2} {3,3,3,3} {5,3,3,2,2,1} I can find the number of edges using the handshaking lemma But how do I construct a graph just given these ...
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How Do I Compute the Eigenvalues of a Small Matrix?

If I have a $2\times 2$ or $3\times 3$ matrix, how should I go about computing the eigenvalues and eigenvectors of the matrix? NB: I am making this question to provide a unified answer to questions ...
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Is the following matrix invertible?

$$\begin{bmatrix} 1235 &2344 &1234 &1990\\ 2124 & 4123& 1990& 3026 \\ 1230 &1234 &9095 &1230\\ 1262 &2312& 2324 &3907 \end{bmatrix}$$ Clearly its ...
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Coordinate-free proof of $\operatorname{Tr}(AB)=\operatorname{Tr}(BA)$?

I am searching for a short coordinate-free proof of $\operatorname{Tr}(AB)=\operatorname{Tr}(BA)$ for linear operators $A$, $B$ between finite dimensional vector spaces of the same dimension. The ...
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What is the usefulness of matrices?

I have matrices for my syllabus but I don't know where they find their use. I even asked my teacher but she also has no answer. Can anyone please tell me where they are used? And please also give me ...
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Determinant of a non-square matrix

I wrote an answer to this question based on determinants, but subsequently deleted it because the OP is interested in non-square matrices, which effectively blocks the use of determinants and thereby ...
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Fast(est) and intuitive ways to look at matrix multiplication?

Most of the time I see matrix multiplication presented and defined, as a seemingly arbitrary sequence of operations. For example, the textbook I'm currently reading for a linear algebra course defines ...
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$I-AB$ be invertible $\Leftrightarrow$ $I-BA$ is invertible [duplicate]

assume $A,B\in M_n(F)$ if $I-AB$ be invertible then how to prove $I-BA$ is invertible and how find inverse of $I-BA$ Thanks in advance
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Why does the inverse of the Hilbert matrix have integer entries?

Let $A$ be the $n\times n$ matrix given by $A_{ij}=\frac{1}{i + j - 1}$. I need to show that $A$ is invertible and the inverse has integer entries. I was able to show that $A$ is invertible. How do I ...
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Cross product: matrix transformation identity

How can one prove the following identity of the cross product? $$(M a)\times (M b)=\det(M) (M^{\rm T})^{-1}(a\times b)$$ $a$ and $b$ are 3-vectors, and $M$ is an invertible real 3x3 matrix.