**31.989 matrices questions.**

$$\left[\begin{smallmatrix}
19&1&1&1&1&1&1&1&1&1&1&1&1&1&1&1&1&1&1&1\\
1&19&1&1&1&1&1&1&1&...

Let $A\in GL_n(\mathbb R)$, $f\in C^2(\mathbb R^n)$. Let $g(x)=f(Ax)$ and $A=(a_{ij})$. Is
$$\frac {\partial g(x)}{\partial x_j}=\sum_{i=1}^{n}a_{ij}f_i(Ax)$$ where $f_i$ denotes the $i$-th partial ...

Prove that if $A$ has linearly dependent rows or columns then $\det(A)=0$.
$$\DeclareMathOperator{\sgn}{sgn}
\det(A)=\sum_{\sigma\in s_n}\Bigl(\sgn(\sigma)\prod_{i=1}^{n}a_{i,\sigma(i)}\Bigr)=\sum_{\...

Volume of a Hypercube After Non-Linear Transformation
The volume of a hypercube can be derived from the determinant of the matrix with rows formed from the difference between one vertex of the ...

How to prove that the following conditions are necessary and sufficient for checking the positive semidefiniteness of a $2 \times 2$ matrix $A$?
$$ a_{11}\geq 0, \qquad a_{22}\geq 0, \qquad a_{11}a_{...

Given an invertible matrix $A \in \mathbb{R}^{n \times n}$ and its polar decomposition
$$
A = U H,
$$
where $U$ is unitary, and $H$ is positive definite. I'm interested in the distance of
$$
arg ( \...

I'm struggling to get the following recurrence relation into a closed form if possible:
$$f(m,0)=f(0,n)=2$$
$$f(0,0)=0$$
$$f(m,n)=f(m-1,n)\cdot(2n+1)(2n+2) + f(m,n-1)\cdot(2m+1)(2m+2)$$
where $m$ ...

Can the matrix transpose be represented by $X^T = AXB$ for a given $A$ and $B$?
I think it is possible, but please correct me if I am wrong--please see my attempt below.
My thinking (not sure ...

From Eq. 51 of the matrix cookbook we know that
$\frac{\partial \log\det (AXB)}{\partial X} = (X^{-1})^\top$,
where $\det(X)$ is the determinant of $X$.
I was wondering what is the derivative of
$\...

I'm struggling to prove/disprove this notion. I've figured if such matrix exists, it has to be nilpotent and non-invertible, and the sum of its eigenvalues is 0. Can anyone chip in?
Edit : Oh yeah, $...

Assume I want to the convergence of a sequence of matrices (finite dimension).
Can I assume that the norm that I am working with is submultiplicative and the generalise with any norms saying that ...

Given the implicit function $p(z,t) = det P(z,t) = 0$, where
$$
P(z,t) = D_m(z) - A(t)
$$
is a polynomial matrix with $D_m(z) = diag(z^{m_1},z^{m_2},\dots,z^{m_N})$ and $A(t) \in \mathbb{C}^{N \times ...

I am asked to prove that for a Hermitian matrix with all eigenvalues negative, the inverse is given by
$$A^{-1} = - \int_0^\infty e^{tA} {\text { d}}t.$$
(I have the corrected the missing minus sign ...

In the book of Linear Algebra by Werner Greub, whenever we choose a field for our vector spaces, we always choose an arbitrary field $F$ of characteristic zero, but to understand the importance of the ...

I have a normal distribution with zero mean and unit variance. I then generated a 100 dimensional vector v1, each element number is from the distribution.
Now I generated another 9 vectors: v2, v3, .....

We know the Pauli spin matrices (along with identity matrix) are a basis for the C2 vector space. All of them are self adjoint.
However, since they are a basis for the entire C2 operator space, linear ...

I know how basic operations are performed on matrices, I can do transformations, find inverses, etc. But now that I think about it, I actually don't "understand" or know what I've been doing all this ...

Suppose A is a symmetric 0-1 matrix. When is $A$ positive semidefinite?

If $A \in \mathbb{C}^{m \times n}$ Show That $\operatorname{Range}(A^*A) = \operatorname{Range}(A^*)$ where * indicates conjugate transpose.

I first saw a proof for the Leibniz formula for computing determinants when I was learning about tensors and the exterior product at college (a "proof" considering that the definition of determinant ...

Suppose that $A$ is an invertible matrix and let $X$ be such that $AX+XA=2I$. Does this imply that $X=A^{-1}$?
I have tried simple algebra manipulations but I have not been able to conclude. For a ...

What is a complete exhaustive classification of all geometric transforms on the $\mathbb{R}^2$ plane obtained with:
2x2 matrices$$A = \pmatrix{a & b \\ c & d}$$ applied to a point $X= (x, y)$....

I have a Hermitian matrix of size $n$, $H_{n\times n}$ with only off diagonal entries, actually all the entries are real in $H$ (very special case). Now my job is to find the anti-commuting matrix say ...

Consider a matrix $A\in\mathbb{R}^{n\times m}$ and another matrix, termed the index matrix, $B\in\{0,1\}^{n\times m}$.
What is the appropriate notation for a submatrix of $A$, call it $C$, built ...

I need to calculate the dominant eigenvalue of the square matrix $\begin{pmatrix}
15 & -4 & -3\\
-10 & 12 & -6\\
-20 & 4 & -2
\end{pmatrix}$ and also the corresponding eigen ...

If A and B are two nxn matrices, how would I prove that rank(AB)$\leq$ rank(A)?

For all $k\in\mathbb{N}$, let $x_k,\ y_k\in\mathbb{R}^n$, and assume that
\begin{equation}
y_{k+1}=\left(I_n-\frac{ax_kx_k^{\rm T}}{b+\|x_k\|^2}\right)y_k,~~~~~~~~~~~~(1)
\end{equation}
where $a\...

I need to find a new base where
X1X2+X2X3+X3X4
Is the sum of squares( diagonal matrix)
Im trying with Lagrange method but without success (row and column actions also accepted)
If anyone can ...

I got the transformation matrix
$$
^aT_b = \begin{pmatrix}
\dfrac{1}{\sqrt2} & -\dfrac{1}{\sqrt2} & 0 & r_x \\
0 & 0 & -1 & 0 \\
\dfrac{1}{\sqrt2} & \dfrac{...

"For $n≥4$, let $U=\{X∈ M_n(\mathbb{C}) | \operatorname{adj}X=I\}$. Which of the following statements are true/false? Justify:
$U$ contains $n-1$ elements,
$U$ contains only scalar matrices,
if $X_1,...

I have started studying linear algebra and I came across determinants. I know there are all sorts of formulae for computing it but I can't find what exactly it is. Why are they calculated the way ...

I have got a question concerning the roots of diagonalizable matrices with integer entries. I know that given a diagonalizable matrix I can compute easily a $k$-th root by computing its diagonal ...

Let $\mathcal{B}=\{v_1, v_2\}$ be a basis of $\mathbb{R^2}$ such that $v_1 =(3,1)$ and $v_2 =(1,1)$.
Let $g:\mathbb{R^2} \times \mathbb{R^2} \to \mathbb{R}$ a dot product such that:
$g(v_1,v_1)=2$
...

Suppose that we have two different discreet signal vectors of $N^\text{th}$ dimension, namely $\mathbf{x}[i]$ and $\mathbf{y}[i]$, each one having a total of $M$ set of samples/vectors.
$\mathbf{x}[m]...

If $A$ is not diagonalizable, then $A^2$ is not diagonalizable or $A$ is singular.
I don't know if I'm right but $A^2=A$. So if $A$ is not diagonalizable then $A^2$ is not diagonalizable. Also If $A$ ...

How does finding out if the null space has only the zero vector prove one-to-one?
One-to-one means that there are distinct images for each distinct vector input.
$$\mathbb R^n \to \mathbb R^m$$
For ...

how do I find matrix 3x3 which is non-diagonalizable and fits this function
$$A^{-1}=A^2+A-I$$
where I is the identity matrix.
I started by multiplying it by $A$ so
$$I=A^3+A^2-A$$
and finally
$$...

Consider $F: \mathbb{R^3} \to \mathbb{R^3}$ represented by:
$ A=
\begin{bmatrix}
1 & 1 & 2 \\
-2 & 5 & 6 \\
1 & -2 & -2 \\
\end{bmatrix}
$
, eigenvalues: $...

There exist stochastic matrices Q such that there is no stochastic matrix P such that $P^2=Q$.
I am interested in the following problem:
For a given stochastic matrix $Q$,
find the stochastic ...

Can I find a solution for $C_{n\times n}$, explicitly, for the given $A_{n\times n}$ and $B_{n\times n}$ such that
$AA^{T} + BB^{T} = CC^{T}$? Here $A^{T}$ denotes the transpose of $A$ and all the ...

I am trying to understand whether or not the product of two positive semidefinite matrices is also positive semidefinite. This topic has already been discussed in the past here. For me $A$ is positive ...

So I am trying to diagonalize this matrix
$$
A=\begin{pmatrix}2&0&\!\!-2\\1&3&\;2\\0&0&\;3\end{pmatrix}
$$
so that those are the rows of the matrix. I know the eigen values ...

Reading an article (A. Veneziani, T. Pereira, A note on the volume form in normal matrix space) I didn't understand this sentence:
"Given a unitary matrix $U$ we may use the representation $U=e^W$ ...

In this article, at the end of the first page, the author claims the following:
Lets take n rows with length m, where each row either contains all 1s or 0s, and then connect them to make a $n\times m$...

What's the best way to interpret Jordan Normal Form (e.g. in terms of a linear map)?
For instance, how should we interpret those $1$'s?

Prove the equivalence of the matrix $2$-norm and $\infty$-norm. That is, for $A \in \mathbb{C}^{m \times n}$ prove that there exist constants $c_1,c_2$ such that $c_1\|A\|_\infty \leq \|A\|_2 \leq c_2\...

The question seems trivial which is why I have some trouble coming up with a proof that is mathematically correct. BTW I cannot yet use eigenvalues as we have not yet covered them in class.
If
$$D=[...

I have a two matrices, $A$ and $B$ with dimensions ($m$, $n$) and ($n$, $m$), respectively. I have an iterative machine learning algorithm (similar to a restricted Boltzmann machine) that changes the ...

If $A^4$ has an eigenvalue $x$, say, does $A$ have an eigenvalue $y$ s.t $y^4=x$?
I do not think this is true for all complex matrices but I cannot seem to find an example…
Also, how do I prove that ...

- linear-algebra
- eigenvalues-eigenvectors
- determinant
- matrix-equations
- vector-spaces
- linear-transformations
- inverse
- abstract-algebra
- matrix-calculus
- norm
- matrix-decomposition
- matrix-rank
- diagonalization
- vectors
- numerical-linear-algebra
- derivatives
- group-theory
- calculus
- optimization
- numerical-methods
- systems-of-equations
- inequality
- transformation
- rotations
- polynomials

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