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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2
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1answer
80 views

Adjoint of a Matrix Definition

Tom M. Apostol in his book "calculus Vol. 2" page 122 (see image below) defines adjoint of a matrix as the transpose of the conjugate of the matrix. Is this definition always correct ? Does it agree ...
1
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1answer
118 views

Order of some matrices in $GL(2,p)$ is coprime with $p$

Let $M$ belongs to $GL(2,p)$ where $p$ is a prime number, and $\det M$ generate $GL(1,p)$, so I want to prove that the order of $M$ is coprime to $p$. I think if $M^{np}=I_2$ that means ...
5
votes
1answer
113 views

What is a $0\times0$ or $0\times3$ matrix?

In the comments to another question, the following exchange was noted: ... wait until you see a 0×0 matrix. and ... or worse, a 0×3 matrix! What are these things? Do they have a name or ...
3
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3answers
57 views

A question regarding $\,3 \times 4$ matrices

Good day, I'm currently studying for an exam and need to learn about matrices too. Well, since I'm not good at English I'll just write what I've done so far. Below is a photo showing the full sheet of ...
1
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1answer
65 views

Solving a Gaussian elimination problem.

I have been given a graph with n nodes. Now, I have to color every node of this graph by k colors, number from 0 to k-1. Now, there is a rule. For a node $x$ with adjacent nodes $y_1 , y_2, y_3, ...
0
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1answer
42 views

Calculating the Basis of a matrix

$$ A= \begin{bmatrix} 1 & 0 & -1 & 7 \\ 0 & -1 & 0 & 1\\ 2 & 1 & 1 & 3 \\ 1 & 0 & -4 & 0 \end{bmatrix} \\$$ I m ...
3
votes
4answers
810 views

If $A^2+A=0$,then $\lambda=1$ cannot be an eigenvalue of A.

Prove the following statement: If $A^2+A=0$,then $\lambda=1$ cannot be an eigenvalue of A. I've been struggling on this question for a couple of hours and don't know how to approach it.
2
votes
1answer
144 views

Induction powers of a matrix

I'm trying to prove that if $A = AB-BA$, where $A,B$ are squared matrices, then $$kA^k = A^kB-BA^k$$ for all $k$ in naturals. I proceed by induction, but I can't arrange the expressions to conclude. ...
14
votes
4answers
1k views

Is $A + A^{-1}$ always invertible?

Let $A$ be an invertible matrix. Then is $A + A^{-1}$ invertible for any $A$? I have a hunch that it's false, but can't really find a way to prove it. If you give a counterexample, could you please ...
1
vote
1answer
188 views

Square roots and unitary matrices

Why is it that for any non-negative matrix $M$ and unitary matrix $U$, we have $$\sqrt{UMU^\dagger}=U\sqrt{M}U^\dagger$$? This question has to do with Problem 2c from this sheet. I think I am ...
0
votes
1answer
31 views

Solve NC + BN =F

I asked this in the computer section; someone suggested asking in the maths section: Is there a simple way to solve the following matrix equation for N: NC + BN = F The matrices B, C, and F are ...
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0answers
35 views

Decomposition of a symmetric semi-definite matrix into sums of sparse symmetric semi-definite matrix

I'll first provide the background. Let $x\in\mathbb{R}^N$ be decomposed into $n$ non-overlapping blocks of variables $x^{(1)},\ldots,x^{(n)}$. We say that $f:\mathbb{R}^N\rightarrow\mathbb{R}$ is ...
2
votes
1answer
115 views

An equivalent condition for a real matrix to be skew-symmetric

$A$ is an $n \times n$ real matrix. prove that $$A=-A^T \iff AA^T=-A^2$$. Thanks.
1
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1answer
125 views

Solution to this linear equation system

So this is my homework : Let $$ A= \begin{bmatrix} 1 & 0 & 1 & 3 \\ 2 & 0 & \lambda & 6 \\ 1 & 1 & 1 & 1 \\ \end{bmatrix} ...
4
votes
3answers
124 views

Map to symmetric matrices is surjective.

Let $M_{k,n}$ be the set of all $k\times n$ matrices, $S_k$ be the set of all symmetric $k\times k$ matrices, and $I_k$ the identity $k\times k$ matrix. Suppose $A\in M_{k,n}$ is such that ...
0
votes
2answers
418 views

Proof or disprove: if an $n \times n$ matrix $A$ is not invertible, then for every $n \times n$ matrix $B$, $AB$ is not invertible.

Proof or disprove: if an $n \times n$ matrix $A$ is not invertible, then for every $n \times n$ matrix $B$, $AB$ is not invertible. Having trouble with this proof. Don't know how to start or what ...
4
votes
1answer
43 views

Computing derivative of function between matrices

Let $M_{k,n}$ be the set of all $k\times n$ matrices, $S_k$ be the set of all symmetric $k\times k$ matrices, and $I_k$ the identity $k\times k$ matrix. Let $\phi:M_{k,n}\rightarrow S_k$ be the map ...
3
votes
1answer
48 views

Derivative of function between sets of matrices

Let $M_{k,n}$ be the set of all $k\times n$ matrices, $S_k$ be the set of all symmetric $k\times k$ matrices, and $I_k$ the identity $k\times k$ matrix. Let $\phi:M_{k,n}\rightarrow S_k$ be the map ...
1
vote
1answer
337 views

For square matrix, right or left inverse is equivalent to inverse. [duplicate]

Definitions: Let $A$ be an $n\times n$ matrix. The $n\times n$ matrix $B(=A^{-1})$ is an inverse for $A$ if $AB=BA=I$. Let $A$ be a $k\times n$ matrix.The $n\times k$ matrix $B$ is a ...
5
votes
1answer
97 views

Determinants and cofactors?

My professor gave us this definition for determinants for a $n \times n$ matrix $A$: $$\det(A) = a_{11}C_{11} + a_{12}C_{12} ... + a_{1n}C_{1n} $$ where $C_{1j}$ is the cofactor of $A$ on $a_{ij}$. ...
5
votes
1answer
239 views

$QR$ decomposition of rectangular block matrix

So I am running an iterative algorithm. I have matrix $W$ of dimensions $n\times p$ which is fixed for every iteration and matrix $\sqrt{3\rho} \boldsymbol{I}$ of dimension $p\times p$ where the ...
2
votes
4answers
106 views

Non-negative, real matrix $\Rightarrow$ non-negative, real eigenvalues?

Does a matrix with all non-negative, real entries have all non-negative, real eigenvalues? Where might I find a proof of such? Ideas: Perhaps we can multiply a prospective eigenvector so its biggest ...
4
votes
3answers
4k views

How to find a transformation matrix, given coordinates of two triangles in $R^2$

I am an undergraduate student, and today I was given two triangles, $T_1$ (green) and $T_2$ (blue) in $R^2$: I was then asked to find the transformation matrix transforming $T_1$ to $T_2$. What I ...
2
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0answers
68 views

homework: rings, matrices and polynomials

$A,B$ are both $n \times n$ and diagonal matrices. Prove that there is a matrix $X$ which is $n \times n$, and polynomials $p$ and $q$ such that $A= p(X), B= q(X)$ Is this true for ANY 2 matrices (we ...
1
vote
1answer
323 views

Condition Number of a block Matrix

Is this hypothesis true? $$cond([A,B])≤cond(A)+cond(B)$$ where $cond$ is the Condition Number. And is this true for rectangular matrices($nxm$)? Let's consider $3$ different conditions for $A$ and ...
0
votes
1answer
172 views

Odd or even row dot product based on order of Matrix

For a given NxN matrix what is not the value of N such that the dot product of a row with itself will be even and dot product of any row with a different row is odd. a) 2k b) 2k,4k,4k+1 c) 4k
1
vote
2answers
446 views

Pre-multiplying and post-multiplying matrices give the same diagonal elements?

If $$X = \left[ \begin{array}{ccc} 3 & 4 & 1\\ 4 & 1 & 3\\ 1 & 3 & 4\end{array} \right]$$ find the possible matrix $Y$ such that: $$XY - YX = I$$ The method my ...
1
vote
1answer
94 views

Some basic questions about matrix rings and reversibility.

Neither commutative rings nor division rings are viable approaches to studying rings of matrices. However, there is a very cool notion of a reversible ring, which looks like it can fill this void. I ...
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0answers
66 views

Rank of free abelian subgroups in integral linear groups

What is the maximal rank of a free abelian subgroup in $\mathrm{GL}(n,\mathbb{Z})$?
1
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1answer
189 views

If $M$ is positive definite, then $\operatorname{det}{(M)}\leq \prod_i m_{ii}$

In the Wikipedia article on positive definite matrices they claim that if $M$ is positive definite, then the determinant of $M$ is bounded by the product of its diagonal entries. How might we show ...
3
votes
1answer
85 views

Cholesky decomposition: any theoretical value?

Just read the Wikipedia article on Cholesky decomposition. All the applications listed there were numerical. Are there theoretical arguments to which it is important? For instance, here there is an ...
1
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1answer
35 views

Convolution composed with an invertible matrix

Let $T$ be an invertible $n \times n$ matrix and let $(h \circ T)(x)$ mean $h(Tx)$. Take functions $f,g$. Does it hold that $(f*g) \circ T = |det(T)| (f \circ T) * (g\circ T)?$ I have had some ...
1
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1answer
340 views

Lower bound for the trace of product of two symmetric matrices

i am stuck on finding a lower bound of $tr(XY)$ of two symmetric matrices in $M_{n}(\mathbb{R})$. I know that it holds $tr(XY)=tr(YX)$ and thus $tr(XY-YX)=0$ and i can remember, that XY-YX is also ...
0
votes
1answer
73 views

How to prove this matrix identity?

This was a single step in a derivation, so I'm assuming there is a way to "see" this without writing down the expression for each entry: $$\sum\limits_{i=1}^n \left(x_i-\frac{1}{n}\sum\limits_{i=1}^n ...
0
votes
1answer
22 views

Applying prices to augmented matrices

The question is as follows (translated): A company wants to rent 20 buses. These 2 buses are to hold 1000 people. They can choose between 3 types, 30, 40 and 60 man buses. How many of each kind ...
1
vote
4answers
155 views

Differentiating second order term of Taylor polynomial (multivariable)

I am trying to derive Newton step in an iterative optimization. I know the step is: $$\Delta x=-H^{-1}g$$ where H is Hessian and $g$ is gradient of a vector function $f(x)$ at $x$. I also know the ...
-3
votes
1answer
58 views

Decomposition of a unitary matrix

Is anyone aware of a decomposition of a unitary matrix into a product of a special unitary matrix and the rest? Edit: A unitary matrix is a matrix that satisfies $\mathbf{U}\mathbf{U}^\dagger=1$ ...
0
votes
1answer
155 views

inequality on minimum singular values of matrices A, B, and A+B

I saw an inequality on minimum singular values of matrices A, B, and A+B. σmin(A)+σmin(B)≥σmin(A+B) I want to know how can it be proven (if it is true) or what is the correct inequality on σmin(A+B) ...
1
vote
1answer
109 views

Given $A$ and $B$ positive-definite matrices and $Q$ unitary matrix, prove that if $A = BQ$, then $A=B$.

Given $A$ and $B$ positive-definite matrices and $Q$ unitary matrix, prove that if $A = BQ$, then $A=B$. $Q$ is unitary, so $QQ^*=I$ If $A$ and $B$ are positive-definite, than $A=A^*$ and $B=B^*$. ...
1
vote
2answers
199 views

How find this matrix $X$,such $X+X^2+X^3$

let matrix $X\in M_{2}(Z)$,and such $$X+X^2+X^3=\begin{bmatrix} 1&2005\\ 2006&1 \end{bmatrix}$$ Find the matrix $X$ My try: let $$X=\begin{bmatrix} a&b\\ c&d \end{bmatrix}$$ where ...
3
votes
1answer
138 views

What is the relationship between the spectrum of a matrix and its image under a polynomial function?

Clearly if $\lambda$ is an eigenvalue of $A$ then $p(\lambda)$ is an eigenvalue of $p(A)$ where $p$ is a polynomial. And there are cases where $A$ may have eigenvalues other than these. Is there a ...
1
vote
2answers
650 views

Center of the Orthogonal Group and Special Orthogonal Group

How can I prove that the center of $O_n$ is $\pm I_n$? I have that $AM = MA$ for all $M$ in $O_n$. And that $A^{-1} = A^T$, $M^{-1} = M^T$. Then $M = A^{-1}MA = A^{T}MA$. I see that since ...
1
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0answers
143 views

Can I find the Pseudoinverse (Moore-Penrose inverse) just by knowing the one-sided inverses of a matrix?

Consider a matrix such as $B = \begin{bmatrix} 1 & 0 & 2 \\ 0 & 1 & 1 \end{bmatrix}$. I know how to compute the right inverses (or in the case of $m\geq n$ the left inverses) and ...
3
votes
3answers
194 views

Chain rule for matrix exponentials

I need help in proving the following theorem: If $M(t)$ is an $n \times n$ matrix of differentiable functions, then $$ \frac{d}{dt}\left( \exp(M(t))\right) = \frac{d}{dt}M(t) \exp(M(t)) = ...
0
votes
2answers
233 views

Rank computation of large matrices

I have to do rank computation of large (million by million) matrices (I don't wish to compute eigenvalues or eigenvectors, just the rank of the matrices). The matrices are sparse. I have been ...
1
vote
0answers
18 views

How transpose of a matrix helps in making better sense of the data

The transpose of a matrix is obtained by flipping it about its diagonal. What is a practical scenario where we gain better insight into a set of data points by transposing it?
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0answers
67 views

Calculating the Jacobian for sampled data

I am reading about the Jacobian matrix which I interpret as a generalized gradient. I would like to take my investigations a bit further, so I have constructed some sample data which I would like to ...
2
votes
1answer
261 views

Show that ${\bf x} \cdot A^t {\bf y} = {\bf y} \cdot A{\bf x}$

Let $A \in \mathcal M_n (R)$ and ${\bf x}, {\bf y} \in R^n$. How can I show that: $${\bf x} \cdot A^t {\bf y} = {\bf y} \cdot A{\bf x} \, ?$$ Thanks for any help.
2
votes
1answer
71 views

What is the smallest Lie subalgebra of $ {{\frak{gl}}_{n}}(\mathbb{R}) $ whose center is the set of $ (n \times n) $-scalar matrices?

We know that the center of the Lie algebra $ {{\frak{gl}}_{n}}(\mathbb{R}) $ of all $ (n \times n) $-matrices is the Lie subalgebra of all $ (n \times n) $-scalar matrices. The Lie algebra $ ...
1
vote
1answer
207 views

Fundamental matrix solution and commutativity.

Please I have a question. Let $$y'(t) = M(t)y(t)~~~~~~~~~~~(*)$$ where $M(t)$ is a matrix with continuous entries on the interval $(a,b)$. Let $Y(t,t_0)$ be its fudamental solution. It is known ...