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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 .

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39 votes
6 answers
31k views

$AB-BA=I$ having no solutions

The following question is from Artin's Algebra. If $A$ and $B$ are two square matrices with real entries, show that $AB-BA=I$ has no solutions. I have no idea on how to tackle this question. I ...
Ishan Banerjee's user avatar
56 votes
4 answers
8k views

Solutions to the matrix equation $\mathbf{AB-BA=I}$ over general fields

Some days ago, I was thinking on a problem, which states that $$AB-BA=I$$ does not have a solution in $M_{n\times n}(\mathbb R)$ and $M_{n\times n}(\mathbb C)$. (Here $M_{n\times n}(\mathbb F)$ ...
Goodarz Mehr's user avatar
  • 2,309
15 votes
2 answers
6k views

What is the inverse of the $\mbox{vec}$ operator?

There is a well known vectorization operator $\mbox{vec}$ in matrix analysis. I've vectorized my matrix equations, did some transformation of vectorized equations and now I want to get back to the ...
Moonwalker's user avatar
9 votes
5 answers
2k views

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&...
Mathematics's user avatar
  • 3,993
22 votes
3 answers
3k views

For every matrix $A\in M_{2}( \mathbb{C}) $ there's $X\in M_{2}( \mathbb{C})$ such that $X^2=A$?

True\False? For every matrix $A\in M_{2}( \mathbb{C}) $ there's $X\in M_{2}( \mathbb{C})$ such that $X^2=A$. I know that every complexed matrix has a Jordan form matrix $J$ such that $P^{-1}CP=J$, ...
user avatar
24 votes
3 answers
7k views

$AB=BA$ implies $AB^T=B^TA$ when $A$ is normal

I am looking for an elementary proof (if such exists) of the following: $$ AB=BA \quad\Longrightarrow\quad AB^T=B^TA, $$ where $A$ and $B$ are $n\times n$ real matrices, and $A$ is a normal matrix, i....
Yiorgos S. Smyrlis's user avatar
8 votes
2 answers
5k views

Is there a unique solution for this quadratic matrix equation?

Here is the quadratic matrix equation I've been looking at lately: $$ Q_{r,r}=A_{r,r}X_{r,r}^2+B_{r,r}X_{r,r}+C_{r,r}=0_{r,r} $$ Note that $A, B, C,$ and $X$ are $r \times r$ matrices. $A$ contains ...
000's user avatar
  • 5,760
5 votes
1 answer
173 views

Matrix equation with symmetric and positive definite matrices [duplicate]

Let matrices $P_0$ and $P_1$ be symmetric and positive definite. Is it possible to find a symmetric matrix $S$ such that $SP_0S=P_1$? If $P_0=I$, this is always possible: $S=\sqrt{P_1}$.
Pete's user avatar
  • 405
19 votes
5 answers
11k views

Find the inverse of a submatrix of a given matrix

I have a $A$ matrix $4 \times 4$. By delete the first row and first column of $A$, we have a matrix $B$ with sizes $3 \times 3$. Assume that I have the result of invertible A that denote $A^{-1}$ ...
John's user avatar
  • 802
13 votes
1 answer
1k views

Is $\exp:\overline{\mathbb{M}}_n\to\mathbb{M}_n$ injective?

More specific to my problem, this is a variation on Is $\exp:\mathbb{M_n}\to\mathbb{M_n}$ injective? which was promptly answered with a counterexample. Let $\mathbb{M}_n$ be the space of $n\times n$ ...
Jeff Snider's user avatar
  • 2,817
9 votes
1 answer
5k views

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&...
Trajan's user avatar
  • 5,204
5 votes
3 answers
2k views

Solving $AB - BA = C$

Suppose $C$ is an $n\times n$ matrix over complex numbers, with trace $0$. Are there always $n\times n$ matrices $A,B$ such that $AB - BA = C$? (Inspired by a recent question which asked for a trace ...
me.dorgan's user avatar
  • 111
4 votes
3 answers
585 views

Find $C$, if $A=CBC$, where $A$,$B$,$C$ are symmetric matrices.

If $A=CBC$, where $A$,$B$,$C$ are symmetric matrices and $A$,$B$ are given find $C$. $A$,$B$,$C$ are assumed to be real valued and $B$ is positive definite matrix. Does the unique solution always ...
Qbik's user avatar
  • 790
12 votes
3 answers
3k views

Solving linear matrix equations of the form $X = AXA^T + C$

I'm trying to solve this linear matrix equation: $$X = AXA^T + C$$ In particular, $$ X = \begin{bmatrix} 1.5 & 1 \\ -0.7 & 0 \end{bmatrix} X \begin{bmatrix} 1.5 & -0.7 \\ 1 & 0 \end{...
hkBattousai's user avatar
  • 4,573
6 votes
1 answer
6k views

Is there a general form for the derivative of a matrix to a power?

Let $S:Mat(2,2) \rightarrow Mat(2,2)$ be the squaring map $S(A)=A^2$ then $[DS(A)]B=AB+BA$. I was wondering if there was a general form for this solution ($S(A)=A^n$, then $[DS(A)]B =$...). I have ...
user2154420's user avatar
  • 1,441
5 votes
2 answers
796 views

Prove uniqueness of solutions of different OLS matrix cases

Let $D = \{(x_1, y_2), (x_2, y_2), \ldots , (x_n, y_n)\}$ where $x_i \in \mathbb{R}^d$ and $y_i \in \mathbb{R}$. One may use linear regression to predict $y$ as $w^Tx$ for some parameter vector $w \in ...
aye.son's user avatar
  • 105
4 votes
2 answers
731 views

Is there any matrix $2\times 2$ such that $A\neq I$ but $ A^3=I$?

Is there any $2 \times 2$ matrix $A \neq I$ such that $A^3=I$? In my opinion: no. Thank you very much.
Jozef's user avatar
  • 7,110
26 votes
3 answers
3k views

If $\,A^k=0$ and $AB=BA$, then $\,\det(A+B)=\det B$

Assume that the matrices $A,\: B\in \mathbb{R}^{n\times n}$ satisfy $$ A^k=0,\,\, \text{for some $\,k\in \mathbb{Z^+}$}\quad\text{and}\quad AB=BA. $$ Prove that $$\det(A+B)=\det B.$$
Iloveyou's user avatar
  • 2,503
5 votes
3 answers
5k views

Proving the inverse of a matrix as a polynomial of matrices

Suppose $A$ is invertible $n\times n$ matrix. Show that $A^{-1}$ can be written as a polynomial of degree at most $n-1$, i.e. $$A^{-1} = c_{n-1}A^{n-1} + \dots + \ c_{1}A + c_{0}I$$ Are there any ...
clocktower's user avatar
  • 1,457
5 votes
2 answers
578 views

Converse linear quadratic optimal control

It is well known that for a linear time invariant system $$ \dot{x} = A x + B u \tag{1} $$ with $(A, B)$ controllable, there exists a static state feedback $u = -K x$ such that the cost function $$ ...
SampleTime's user avatar
  • 3,466
3 votes
1 answer
2k views

Solving for $A$ in $Ax = b$

How does one solve for the matrix $A$ in the system $$ Ax = b$$ when say, $A$ is a $3\times 3$ matrix and $x$ and $b$ are both $3 \times 1$. $x$ and $b$ are obviously known. The obvious approach ...
Doug's user avatar
  • 33
3 votes
2 answers
694 views

Solving inhomogenous ODE

I have an inhomogenous ODE. The main issue here is variables are matrices. It is bit of matrix calculus. A solution would be highly appreciated interms of x . I guess we can use same methods for ...
Nirvana's user avatar
  • 1,707
1 vote
2 answers
208 views

Matrix with integer coordinates

I am dealing with the test of the OBM (Brasilian Math Olympiad), University level, 2016, phase 2. As I've said at this topic (question 1), this other (question 2) and this (question 3, yet open), I ...
Quiet_waters's user avatar
  • 1,525
1 vote
2 answers
301 views

Let $A, B$ be $n\times n$ with $n\ge 2$ nonsingular matrices with real entries such that $A^{-1} + B^{-1} =(A+B)^{-1}$

Let $A, B$ be $n\times n$ with $n\ge 2$ nonsingular matrices with real entries such that $$A^{-1} + B^{-1} =(A+B)^{-1}$$ then prove that $\operatorname{det}(A)=\operatorname{det}(B)$. Also show ...
MathBS's user avatar
  • 3,124
0 votes
0 answers
434 views

Consider the following systems of equations with unknowns x, y. For which pairs of values (a, b) does each system have an infinite no. of solutions?

For the below questions, I understand how to find the different values of $a$ when looking for a unique solution for each, but I do not understand the second part on pairs of $a$ and $b$ to find ...
Paul's user avatar
  • 1
20 votes
5 answers
31k views

Do row operations change the column space of a matrix?

I know that (i) row operations do not change the row space (ii) column operations do not change the column space and (iii) row rank = column rank (but this is sort of unrelated, I think). But, ...
User001's user avatar
16 votes
2 answers
198k views

what do free variable and leading variables mean?

What do the leading variables and free variables in a matrix mean? I have the system below and am trying to understand which are which. I searched a lot for this, please help me ! $$w + x + y + z = 6 \...
user136980's user avatar
11 votes
4 answers
902 views

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 $A$ ...
shwetha's user avatar
  • 705
10 votes
3 answers
2k views

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 ...
Riley's user avatar
  • 2,757
4 votes
1 answer
2k views

Quadratic matrix equation $XAX=B$ [duplicate]

Let $A$ and $B$ be two positive semidefinite $n \times n$ matrices. Does the following quadratic matrix equation have a solution in the set of real symmetric matrices? $$XAX=B$$ It's a special case ...
Jef's user avatar
  • 83
4 votes
1 answer
5k views

Exact solution of overdetermined linear system

Given a (possibly) overdetermined linear system $Ax=b$, where $A$ is full rank and $A \in \mathbb{R}^{m \times n}, \quad m \ge n$ Does the least squares method provide an exact solution (instead of ...
plasmacel's user avatar
  • 1,272
3 votes
3 answers
1k views

Find all the linear involutions $f: E \to E$, where $E$ is a finite-dimensional real vector space

Can someone help me? I've been thinking about this question for a while and got stuck. At first I only found the Identity transformation ($I$) and the anti-Identity transformation ($-I$). But then I ...
Pedro Amorim's user avatar
2 votes
1 answer
744 views

Solve matrix equation $AXB+CX=D$

How to solve matrix equation $AXB+CX=D$ for $X$? If it is not solvable, are there any numerical methods to do it?
webjunkie's user avatar
2 votes
4 answers
6k views

Show that a matrix $A=\pmatrix{a&b\\c&d}$ satisfies $A^2-(a+d)A+(ad-bc)I=O$

Let $A= \begin{bmatrix} a & b \\ c & d \end{bmatrix} ,a,b,c,d\in\mathbb{R}$ . Prove that every matrix $A$ satisfies the condition $$A^2-(a+d)A+(ad-bc)I=O .$$ ...
user300045's user avatar
  • 3,469
1 vote
3 answers
231 views

Hermitian matrix such that $4M^5+2M^3+M=7I_n$

$n$ is a positive integer. Besides the identity matrix $I_n$, does there exist other $n\times n$ Hermitian matrix $M$, such that the following equality $$4M^5+2M^3+M=7I_n $$ hold? I try this: ...
ziang chen's user avatar
  • 7,791
1 vote
1 answer
934 views

Show that the matrix with a's on the diagonal and b's off the diagonal has got an inverse with the same structure [duplicate]

Let assume that we have got some $a, b \in \mathbb{R}$ such that $a \neq 0$, $a \neq b$ and $A \in M_{n \times n}(\mathbb{R})$ of the form $$ A = \begin{bmatrix} a & b & \cdots & b \\ ...
nadamai's user avatar
  • 13
-1 votes
4 answers
4k views

If the product of two square matrices is invertible, then both matrices are invertible

If $A$ and $B$ are $n\times n$ matrices, and $AB$ is invertible then $A$ and $B$ are invertible. I started out by writing that since $AB$ is invertible, then for the equation $ABx=b$ has a unique ...
user278039's user avatar
14 votes
2 answers
28k views

Solving the quadratic equation for matrices

Suppose that $A,\;B,\;C,\;$and $X$ are all real commuting matrices. I am curious how to solve $$AX^2+BX+C=0$$ for $X$. In addition what properties do we need on $A,\;B,$ and $C$ for the solution to ...
Wintermute's user avatar
  • 3,838
11 votes
3 answers
14k views

Solving non-square linear matrix equations

Let's say we have $${\bf A} = {\bf B} {\bf X}$$ where $\bf A$ and $\bf B$ are known matrices, $\bf X$ is unknown. In case $\bf B$ was square, a solution can be found by ${\bf B}^{-1} {\bf A} = {\bf X}$...
Milo Wielondek's user avatar
9 votes
3 answers
272 views

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 ...
pointer's user avatar
  • 1,811
8 votes
2 answers
2k views

Solving the matrix equation $X^tA+A^tX=0$ for $X$ in terms of $A$

Suppose that I know $A$. And all matrices in the equation are square matrices. I want to solve for $X$ given that $$X^tA + A^tX = 0$$ I'm not really good at matrix calculus. Is it possible to solve ...
stressed out's user avatar
  • 8,180
7 votes
2 answers
297 views

For $a_i \in \mathbb{C}$, does $(1+a_1^k)(1+a_2^k)\cdots (1+a_n^k)=1$ for any positive integer $k$ imply $a_1=\cdots =a_n =0$?

For $a_i \in \mathbb{C}$, if $(1+a_1^k)(1+a_2^k)\cdots (1+a_n^k)=1$ holds for all positive integers $k$, does it follow that $a_1=\cdots =a_n =0$? In fact, I want to prove $|I+A^k|=1$ for any positive ...
levi's user avatar
  • 99
7 votes
2 answers
12k views

Simultaneously diagonalization of two matrices.

Let $A$ be a real symmetric matrix and $B$ a real positive-definite matrix. Is it possible to simultaneously diagonalize of $A$ and $B$? Thank you very much.
LJR's user avatar
  • 14.6k
6 votes
2 answers
247 views

Prove that for every complex $2$x$2$ matrix $A$, there exists a matrix $X$ such that $X^3=A^2$.

This is a problem I've been stuck on for a while. I know that this statement is false if it had $A$ instead of $A^2$, for example, the matrix $A = \begin{bmatrix} 0&1\\0&0 \end{bmatrix}$ can't ...
user402016's user avatar
6 votes
3 answers
2k views

Solving matrix equation $XA=AY$ with known $X$ and $Y$

I am having problem in solving set of matrix multiplication. There are three matrices $A,X$ and $Y$, all are non-singular $2\times 2$ matrices. Where matrix $X$ and $Y$ are known and $A$ is unknown. $...
Salman's user avatar
  • 166
6 votes
4 answers
3k views

Proof if $I+AB$ invertible then $I+BA$ invertible and $(I+BA)^{-1}=I-B(I+AB)^{-1}A$ [duplicate]

I have the following question : Proof if $I+AB$ invertible then $I+BA$ invertible and $(I+BA)^{-1}=I-B(I+AB)^{-1}A$ I managed to proof that $I+BA$ invertible My proof : We know that $AB$ and $BA$ ...
JaVaPG's user avatar
  • 2,726
6 votes
1 answer
3k views

Tangent Space of SL(n,R) at arbitrary point, e.g. not at $\mathbb{1}$

I am looking for the tangent space of $SL(n,\mathbb{R}) = \{A\in\mathbb{R}^{n\times n} | \det{A}=1 \}$ (actually $n=3$) at an arbitrary $M\in SL(n,\mathbb{R})$. From now on I will omit the $\mathbb{R}$...
rehctawrats's user avatar
5 votes
3 answers
289 views

Computing $(A\otimes I + I \otimes A)^{-1} \text{vec}B$

Suppose I have two positive semi-definite $n$-by-$n$ matrices $A$, $B$ and an $n$-by-$n$ identity matrix $I$, and I'm looking for a way to compute, approximate or bound the following quantity: $$(A\...
Yaroslav Bulatov's user avatar
5 votes
3 answers
2k views

The basis of a matrix representation

If I have the linear map $f:\Bbb{R}^n\rightarrow \Bbb{R}^m$ then we can write $f$ as like the following: $$f\left(\vec x\right)=A\vec x$$ Where $A$ is a matrix. I think $A$ is called the standard ...
user avatar
5 votes
4 answers
9k views

If matrix $AB=A$, does it mean B must be an identity matrix?

If matrix multiplication $AB=A$, does it mean $B$ must be an identity matrix? If not, why? What conditions? $A$ is not a zero matrix.
CyberPlayerOne's user avatar

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