Questions tagged [matrix-equations]

This tag is for questions related to equations, with matrices as coefficients and unknowns. A matrix equation is an equation in which a variable stands for a matrix .

Definition: Let $~v_1,~v_2,~\cdots~,v_n~ $ and $~b~$ be vectors in $~\mathbb{R^n}~$. Consider the vector equation $$x_1~v_1+x_2~v_2+~\cdots~+x_n~v_n=b~$$This is equivalent to the matrix equation$$~Ax=b~$$

where $~~A=\begin{pmatrix} \cdot & \cdot & \cdots & \cdot \\ v_1 & v_2 & \cdots & v_n \\ \cdot & \cdot & \cdots & \cdot \\ \end{pmatrix};~~ x=\begin{pmatrix} x_1 \\ x_2\\ \cdots\\ x_n \end{pmatrix} ~~\text{and}~~ b=\begin{pmatrix} b_1 \\ b_2\\ \cdots\\ b_n \end{pmatrix}$

Since a matrix equation $ ~AX=B~$ (where $ ~X~$ is a column vector of variables) is equivalent to a system of linear equations, we can use the same methods we have used on systems of linear equations to solve matrix equations. Namely:

$(1.)~~$ Write down the augmented matrix $ ~A \vdots B$.

$(2.)~~$ Row-reduce to a new augmented matrix $~ \overline A \vdots \overline B~$ in row echelon form.

$(3.)~~$ Use this new matrix to write a matrix equation equivalent to the original one.

$(4.)~~$ Use this new, equivalent matrix equation to find the solutions to the original equation.

In mathematics, matrix equation (which is a system of linear equations) is the basis and a fundamental part of linear algebra, a subject which is used in most parts of modern mathematics. Computational algorithms for finding the solutions are an important part of numerical linear algebra, and play a prominent role in engineering, physics, chemistry, computer science, and economics. A system of non-linear equations can often be approximated by a linear system (see linearization), a helpful technique when making a mathematical model or computer simulation of a relatively complex system.

Very often, the coefficients of the equations are real or complex numbers and the solutions are searched in the same set of numbers, but the theory and the algorithms apply for coefficients and solutions in any field. For solutions in an integral domain like the ring of the integers, or in other algebraic structures, other theories have been developed, see Linear equation over a ring. Integer linear programming is a collection of methods for finding the "best" integer solution (when there are many). Gröbner basis theory provides algorithms when coefficients and unknowns are polynomials. Also tropical geometry is an example of linear algebra in a more exotic structure.

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If square matrices $A^2 + B^2 = 2AB$, then prove that $p_A(x) = p_B(x)$

Original problem statement: Let $A, B \in M_n(\mathbb{C})$ such that $A^2 + B^2 = 2AB$. Prove that for any $x \in \mathbb{C}$: $$det(A - xI_n) = det(B-xI_n)$$ Now the first observation, the equality that has to be proven is the definition of the…
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A conjecture on the Lyapunov equation

Let $A\in\mathbb{R}^{n\times n}$ be a Hurwitz stable matrix (i.e., all the eigenvalues of $A$ have strictly negative real part). Let $X\in\mathbb{R}^{n\times n}$ be a positive semi-definite matrix of unit trace, that is $X\succeq 0$ s.t.…
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Are there uncountably many $A\in M_3 (\mathbb {R})$ such that $A^8=I $?

I'm working on the following problem: Let $A \in M_3 (\mathbb {R})$ be such that $A^8=I$. Then the minimal polynomial of $A$ can only be of degree $2$. the minimal polynomial of $A$ can only be of degree $3$. either $A = I$ or $ A = -I$. …
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Prove that the Sylvester equation has a unique solution when $A$ and $-B$ share no eigenvalues

We are given the Sylvester equation $AX+XB=C$ with complex matrices. I am trying to understand the proof that if $A$ and $-B$ share no eigenvalues, then there is a unique solution $X$ for any $C$. The proof is on Wikipedia and reads like…
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Solving non square matrix equations

Lets say we have: $\mathbf{A=BX}$ Where A and B are known matrices, X is unknown. In case B was square, a solution can be found by $\mathbf{B^{-1}A=X}$. But how do you attempt to solve for X when B is not square, i.e. $n\neq m$?
Milo Wielondek
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How to rotate the positions of a matrix by 90 degrees

I have a 5x5 matrix of values. I'm looking for a simple formula that I can use to rotate the position of the values (not the values themselves) 90 degrees within the matrix. For example, here is the original matrix: 01 02 03 04 05 06 07 08 09 10 11…
deadlydog
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Solutions of $XA=XAX$.

All matrices are real and $n \times n$. The matrix $A$ is given. I am interested in solving $XA=XAX$. In particular, I would like some characterization of matrices that satisfy this equation. For instance, a useful characterization would be: any…
ziutek
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Prove that if the sum of each row of A equals s, then s is an eigenvalue of A.

Consider an $n \times n$ matrix $A$ with the property that the row sums all equal the same number $s$. Show that $s$ is an eigenvalue of $A$. [Hint: Find an eigenvector] My attempt: By definition: $Ax = sx$ which implies that $(A - sI)x = 0$ $s$ is…
user3672888
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Existence of some type matrix

Is there square matrix $A$ of size $3$ with real entries such that $$ \operatorname{tr}(A)=0\text{ and }A^2+A^T=I. $$ I have proved that there is not with size $2$ using definition of "trace", but for size $3$ it becomes complicated. Here is the…
pointer
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Method of characteristics for a system of pdes

I can do parts a) and b) as follows $\begin{pmatrix} 1&0&0 \\ 0&1&0 \\ 0&0&1\end{pmatrix}\frac{\partial}{\partial{}x}\begin{pmatrix} u \\ v \\ w\end{pmatrix}+\begin{pmatrix} 1&1&0 \\ 1&2&1 \\ 0&1&1\end{pmatrix}\frac{\partial}{\partial…
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A linear algebra problem regarding $AB-BA=A$

Let $A$ be a $n \times n$ complex matrix. Show (1) is equivalent to (2) (1) There exists $B$ such that $AB-BA = A$ (2) $A^n=0$ Furthermore, prove $B^n\neq 0$ in (1) if $A \neq 0$. Attmept for (1) $\implies$ (2) $A^2 = A^2B - ABA = ABA-BA^2$.…
Kim
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Find $B=A^2+A$ knowing that $A^3=\begin{bmatrix}4&3\\-3&-2\end{bmatrix}$

Find $B=A^2+A$ knowing that $A^3=\begin{bmatrix}4&3\\-3&-2\end{bmatrix}$ Is there a way to solve this rather than just declaring a matrix $$A=\begin{bmatrix}a&b\\c&d\end{bmatrix}$$ and then trying to solve a system of cubic equations? My…
C. Cristi
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How do you find all solutions to the matrix equation $XAX=A^T$?

I was recently asked to solve a problem in a programming interview involving word squares, and on further reflection I realized it could be recast as a linear algebra question. Since my solution has a worst-case time complexity of $O(n!)$ if $n$ is…
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Prove that matrix can be square of matrix with real entries

Prove that matrix \begin{bmatrix}1&0&0\\0&-1&0\\0&0&-1\end{bmatrix} can be square of matrix with all real entries. I have found one such matrix to be \begin{bmatrix}1&0&0\\0&1&-1\\0&2&-1\end{bmatrix} but is there an elegant way to do it without…
Mathematics
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How can I find all the matrices that commute with this matrix?

I would like to find all the matrices that commute with the following matrix $$A = \begin{pmatrix}2&0&0\\ \:0&2&0\\ \:0&0&3\end{pmatrix}$$ I set $AX = XA$, but still can't find the solutions from the equations.
Rongeegee
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