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
118 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
vote
1answer
141 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
130 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
427 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
45 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
387 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
101 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
271 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
109 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
votes
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
387 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
212 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
526 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
100 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 ...
1
vote
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
vote
1answer
206 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
88 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
vote
1answer
36 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
vote
1answer
397 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
74 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
175 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
62 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
201 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
110 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
204 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
140 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
726 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
vote
0answers
158 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 ...
4
votes
3answers
217 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
279 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
19 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?
0
votes
0answers
71 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
267 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
226 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 ...
0
votes
1answer
49 views

Representing complex numbers as matrices, show that $A(z)+A(z')=A(z+z')$

I am doing a task where in which I am representing complex numbers as matrices, so $z=x+iy \in \Bbb C$ is represented by: $A(z)=\begin{bmatrix} x & -y \\ y & x \end{bmatrix}$ Now I have to ...
2
votes
2answers
58 views

Singular Value Decomposition-noisy data

I have a system of the form $$Ay=f,$$ where $A$ is a $N\times4$ matrix, $y$ a 4-element array of unknows and $f$ an $N$-element array. I add Gaussian noise in my data. I tested the following ...
0
votes
1answer
102 views

Need to find N value where each sum A+B is different

I need to find N value (in this case 12, but next time they could more o less) and I need that every sum of two value is a unique number. In the picture below you can see an easy matrix where there ...
3
votes
1answer
101 views

Proving that in this matrix, if the product of each column is the same, so is the product of each row

Consider an $n \times n$ matrix of $2n$ distinct numbers $a_1,a_2, \dots, a_n, b_1, b_2, \dots, b_n$, such that the number at the intersection of the $i$th row and the $j$th column is $a_i + b_j$: $$ ...
1
vote
2answers
8k views

What are the applications of matrices in real world?

Matrices are considered very important in mathematics. What are some examples of applications of matrices to real world problems that would be understandable by a layman?
2
votes
2answers
56 views

What is this good for - determinants

Ok, Using RRef and the identity matrix I can find the inverse matrix and the solution vector with out (directly) finding the determinant of a square matrix. But I have to believe, if this was the only ...
1
vote
1answer
324 views

Tensors = matrices + covariance/contravariance?

I have read several topics on tensors but it is still not clear to me. Tensors are different from matrices because they contain additional information about how do they transform. To fully specify a ...
1
vote
1answer
472 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 ...
1
vote
1answer
127 views

Equality involving norm on Cholesky/QR factorization

Let $B$ be symmetric and positive definite and $B=B^{\frac{1}{2}}B^{\frac{1}{2}}$ the Cholesky factorization. Having $A=QR$, why can we follow the last equality in the following? $$ || A ...
2
votes
1answer
30 views

Question on Matrix Transform Operations

Given $e = Y - XB$, where $ e = \begin{bmatrix} e_1 \\ \vdots \\ e_n \\ \end{bmatrix} $, $ Y= \begin{bmatrix} y_1 \\ \vdots \\ y_n \\ \end{bmatrix} $, $ X= \begin{bmatrix} 1 & ...
0
votes
1answer
22 views

Inverse of $\{a_1 A_1,…,a_n A_n\}$

$a_1,...,a_n\in \mathbb{R}$ $A_1,...,A_n$ are the rows of the invertible matrix A I am trying to find a regular formula for this. Is it possible? Thanks for help!
0
votes
2answers
358 views

inequality with the Frobenius norm for matrices

Let $A\in M_n$. How can I show that $$\left|{\textrm{Tr}(A)\over\sqrt{n}}\right|\leq \Vert A\Vert_F$$ I tried it using the Cauchy-Schwarz inequality.