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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 solve matrix equation $AX+XB=C$ for $X$

How does one solve the matrix equation $AX+XB=C$ for $X$? It doesn't seem too difficult. I tried many times but failed. I'm an adult student... I am now vexed about Gilbert Strang - An Introduction to Linear Algebra. I don't even understand a single…
Alfred
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Subtract matrix from scalar

Is this even possible? Since you can subtract on the right-hand side I think there must be a way to do it from left-hand side too. I would like to calculate this: 3 - [2 1] = ??
c0dehunter
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If a matrix is triangular, is there a quicker way to tell if it is can be diagonalized?

I hope it is alright to ask something like this here, I am having trouble keeping up with all the special cases and my book is being kind of vague. I know how to do the standard method of finding diagonal matrices, but I know that triangular…
Luke
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Under what conditions is $AA^T$ invertible?

Given a matrix $A$ with dimensions $m \times n$, is $B=AA^T$ invertible if and only if the rows of $A$ are linearly independent? So far, I've tried writing A as row vectors, $$A = \begin{bmatrix}v_1\\ v_2\\ \vdots \\ v_m\end{bmatrix}$$ where…
Vincent Tjeng
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A question on Gram matrix

The entries of Gram matrix is defined by $ \langle x_i,x_j\rangle$ in the $(i,j)^{\text{th}}$ position. It is known that Gram matrix is positive semidefinite. Is it still positive semidefinite if $ \langle x_i,x_j\rangle$ is replaced by $…
Sunni
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Prove that $MN-NM$ is singular.

Let $M$ and $N$ be square matrices such that $M^2+N^2=MN$. Then prove that $MN-NM$ is singular. So basically I have to prove: $\det(MN-NM)=0$. I tried to prove this by multiplying the given condition by the inverse of matrices $M$ and $N$, but…
Schrodinger
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Prove that $AB=BA=0$ for two idempotent matrices.

Suppose that $A, B$ are idempotent matrices ($A^2=A$), such that $A + B$ is idempotent, prove that $AB = BA = 0$
Chance
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What is the largest determinant you can get by filling in 0,1 or 2 into a 4-by-4 matrix?

For example $$\left| \begin{array}{ccc} 2 & 0 & 0 & 2 \\ 2 & 0 & 2 & 0 \\ 0 & 2 & 1 & 2 \\ 2 & 2 & 0 & 0 \end{array} \right|=40$$ Can it get bigger than that? And what's your approach?
xzhu
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Let $M$ be a non-zero $3\times 3$ matrix satisfying $M^3=O$

Let $M$ be a non-zero $3\times 3$ matrix satisfying $M^3=0$, where $0$ is the $3\times 3$ zero matrix. Then $(A)\det(\frac{1}{2}M^2+M+I)\neq0$ $(B)\det(\frac{1}{2}M^2-M+I)=0$ $(C)\det(\frac{1}{2}M^2+M+I)=0$ $(D)\det(\frac{1}{2}M^2-M+I)\neq0$ This is…
Vinod Kumar Punia
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Is $[X,Y] \neq 0$ the sufficient condition of $e^{X+Y} \neq e^Xe^Y$?

We know that if X commutes with Y, where X and Y are $n\times n$ matrices, then we have $$e^{X+Y}=e^Xe^Y$$ However, can we conclude that $e^{X+Y} \neq e^Xe^Y$ if X doesn't commute with Y ? Is there any counterexample ? Or prove if it is right.
William Huang
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How to prove that an M-matrix is inverse-non-negative?

Wikipedia says that The inverse of any non-singular M-matrix is a non-negative matrix." To be more precise, if $A$ is an M-matrix, then the entries of the inverse of $A$ are all non-negative, i.e. $A^{-1} \geq 0$. How do I prove this result?
I Like to Code
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Smallest Non-negative number in a matrix

There is a question I encountered which said to fill an $N \times N$ matrix such that each entry in the matrix is the smallest non-negative number which does not appear either above the entry or to its left. That is for $N = 6$ the matrix looks…
Anubhav Agarwal
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$M,N\in \Bbb R ^{n\times n}$, show that $e^{(M+N)} = e^{M}e^N$ given $MN=NM$

I am working on the following problem. Let $e^{Mt} = \sum\limits_{k=0}^{\infty} \frac{M^k t^k}{k!}$ where $M$ is an $n\times n$ matrix. Now prove that $$e^{(M+N)} = e^{M}e^N$$ given that $MN=NM$, ie $M$ and $N$ commute. Now the left hand side of…
Slugger
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Show that the identity matrix $I$ must have norm $1$.

I am trying to understand why the identity matrix $I$ must have a norm $1$, for any choice of matrix-norm $|\cdot|$? How would i show this?
user67411
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Square root of a specific 3x3 matrix

From a problem set I'm working on: (Edit 04/11 - I fudged a sign in my matrix...) Let $A(t) \in M_3(\mathbb{R})$ be defined: $$ A(t) = \left( \begin{array}{crc} 1 & 2 & 0 \\ 0 & -1 & 0 \\ t-1 & -2 & t \end{array} \right).$$ For which $t$ does…
Michael Chen
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