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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3answers
52 views

“$1$ is an eigenvalue of $A^n$” implies an eigenvalue of $A$ is a root of unity?

Let $A$ be a square matrix. If $1$ is an eigenvalue of $A^n$, then is it true that there is an eigenvalue of $A$ which is a root of unity?
5
votes
1answer
196 views

$AB-BA=A$ implies $A$ is singular and $A$ is nilpotent. [duplicate]

Let $A$ and $B$ be two real $n\times n$ matrices such that $AB-BA=A$ Prove that $A$ is not invertible and that $A$ is nilpotent. My attempt is the following. It holds that $AB=(B+I)A$ If ...
3
votes
2answers
54 views

Function for diagonalizing a vector.

I was playing around whith the idea of what operation (function) should I perform (apply) over a vector $\mathbf{a} = (a_1,a_2, \ldots, a_N)^T \in \mathbb{R}^N$ to come up with the following matrix: ...
0
votes
1answer
62 views

What is the derivative of a skew symmetric matrix?

I'm trying to work out some Jacobians and I ran across a problem. If I have a function of a vector making it a skew symmetric matrix, like below, what is the derivative $f'$? $$ ...
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2answers
46 views

Method to Multiply many Matrices simultaneously

Imagine you have three random matrices $$ A = \left[ \begin{array}{cc} 1 & 2 \\ 3 & 4 \end{array} \right] ; \quad B = \left[ \begin{array}{cc} 3 & 4 \\ 5 & 6 \end{array} \right]; ...
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vote
2answers
88 views

When is a companion matrix diagonalizable and what does this say about the associated field extension?

Consider the $n\times n$ matrix $$ M=\begin{pmatrix} 0 & 0 & 0 & \cdots & 0 & 0 & -c_0\\ 1 & 0 & 0 & \cdots & 0 & 0 & -c_1\\ 0 & 1 & 0 & ...
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votes
2answers
75 views

Finding the inverse of a matrix by Gaussian elimination

I spent last hours trying to figure out how to solve the inverse matrix to this matrix: $$\begin{pmatrix} 2 &-3 & 1 \\ 1 & 2 &-1 \\ 2 & 1 & 1 \end{pmatrix}$$ The correct ...
9
votes
1answer
113 views

Does the Trace product in a semigroup have any relation with Trace of a matrix / matrix product

I recently read an article on generalized inverses and Green's relations (by X.Mary). The framework is semigroups, but obviously it has a lot of application within matrix theory. In the article ...
2
votes
0answers
68 views

kernel space of linear combination of matrices

Suppose $A$ and $B$ are $N\times N$ matrices so that for every $x$ and $y$, $xA+yB$ has a kernel of dimension at least $2$. Is it necessarily true that $\ker(A)\cap\ker(B)$ has dimension at least ...
1
vote
2answers
311 views

Continuity of the inverse matrix function

For a differentiation module I am taking one of the exercises (not homework) asks: Show that the set $U \subset \mathbb{R}^{n^{2}}$ of matrices $A$ with $det(A) \neq 0$ is open. Let $A^{-1}$ be the ...
0
votes
1answer
69 views

How to calculate a Frobenius norm?

Suppose that $A$ is an $n \times m$ ($n$ less than $m$) full rank matrix. Apply Gram-Schmidt orthogonalization to the rows of $A$, then we get an $n \times m$ matrix $B$ with orthonormal columns. ...
2
votes
0answers
22 views

Find the coordinates of the matrix after a reflection in the given line.

$$\left[\begin{matrix}-8 & 1 & -7\\ -7 & -5 & 1\end{matrix}\right]$$ The given line is the y axis. I cannot show my work for I have no clue on how to solve this.
1
vote
1answer
51 views

Calculating eigenvalues

I need to calculate the characteristic polynomial and eigenvalues of the following matrix. It's been a long time since my linear algebra courses, so I have pretty much lost the ability to compute such ...
2
votes
1answer
103 views

Eigenvalues and Eigenvectors for matrix. Complex Eigenvalues

How can I find out the eigenvectors for this matrix: $$A= \begin{pmatrix} -3 &0&0\\ 0&3&-2\\ 0&1&1 \end{pmatrix} $$ I found the eigenvalues: $\lambda_{1}=-3$, ...
1
vote
3answers
174 views

How to tell if a linear system is consistent

So I have a list of equations and have made it into REF which gives me $$\left[\begin{matrix}1 & -1 & 1 \\ 0 & 1 & -1 \\ 0 & 0 & 0\end{matrix}\middle|\begin{matrix}1 \\ ...
0
votes
0answers
26 views

The second eigenvalue of a reducible stochastic matrix

The magnitude of the second dominant eigenvalue of a reducible matrix, as I know, is supposed to be 1, why it's not the case for this matrix : $$ \begin{matrix} 0 & 1 & 0 ...
0
votes
2answers
35 views

What can you say about the range space and null space

Let $V$ be a vector space over a field $F$ and $T$ a linear operator on $V$. If $T^2$$=$ $0$, what can you say about the relation of the range of $T$ to the null space of $T$?
2
votes
1answer
73 views

How much linearly independent? or linearly dependent?

I want to improve a rank-deficient matrix by augmenting a row vector to it. However, unfortunately, I have only very 'similar' vectors.. For example, my matrix is somewhat like.. \begin{bmatrix} 1 ...
2
votes
1answer
31 views

Spectral norm of a matrix obtained by setting some entries to zero

For example can we say, that if $A$ is original matrix and $A'$ obtained from $A$ by zeroing some elements then $\|A\|_2 \geq \|A'\|_2$?
0
votes
2answers
33 views

Change of Base matrices

Let $E$ and $F$ be two bases of the same n dimensional vector space $U$ $\bullet$ if $P$ is the change of base matrix from $E$ to $F$ and Q the change of base matrix from $F$ to $E$ then ...
1
vote
1answer
41 views

Matrix multiplication: $X_{r \times c}$ and $Y_{c \times d}$

Matrix $X$ has $r$ rows and $c$ columns, and matrix $Y$ has $c$ rows and $d$ columns, where $r, c$, and $d$ are different. Which of the following must be false? The product $YX$ exists The product ...
2
votes
2answers
82 views

How do I prove this matrices question?

Let $$ A = \begin{pmatrix} 1 & 1 & 1 \\ 1 & \omega^2 & \omega\\ 1 & \omega & \omega^2 \end{pmatrix}$$ where $\omega \ne 1$ is a cube root of unity. If ...
2
votes
0answers
30 views

Multiplication of matrices [duplicate]

When we add two matrices we just simply add the corresponding elements but when we multiply two matrices there is a much more complex process.Why does it happens?
0
votes
1answer
44 views

Prove: $R(A+B) \subset R(B)+R(A)$

Prove: $R(A+B) \subset R(B)+R(B)$ If it's not clear $R(A)$ is the the row-space of $A$. Let $(A+B)_i$ the $i$-th row of $(A+B)$. We can write it as a linear combination of $A$ and $B$. ...
2
votes
2answers
105 views

How to find a basis for a tricky 2x 2 matrices vector space

Consider the vector space of 2 x 2 matrices :\begin{bmatrix}a&b\\0&c\end{bmatrix} such that a and c are rational numbers and b is a real number with rational numbers as the field of this ...
1
vote
1answer
144 views

Linear Algebra matrix notation

My question is referring to the following $4 \times 6$ matrix: $$\begin{bmatrix} 0 & 1 & 2 & 0 & 0 & 2 \\ 0 & 0 & 0 & 1 & 0 & 3 \\ 0 & 0 & 0 & 0 ...
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vote
2answers
73 views

Eigenvalues of $A+B$ in this special case

Let $A$ and $B$ are real, square matrices with the same dimension. We know that $\text{rank } A = 1$ and we know the eigenvalues of $A$. Furthermore, we know that $B$ has only zeros in the diagonal, ...
1
vote
1answer
91 views

Hessian matrix as derivative of gradient

From a text: For a real-valued differentiable function $f:\mathbb{R}^n\rightarrow\mathbb{R}$, the Hessian matrix $D^2f(x)$ is the derivative matrix of the vector-valued gradient function $\nabla ...
1
vote
0answers
56 views

Two person zero sum problem, help/guidance needed..

I'm a computer science student and I have this problem I need to solve for my games theory course. I don't have an example to follow, or use as guidance, and my colleagues are not very helpful( as in, ...
0
votes
1answer
17 views

Understanding the basis term

Consider: $$\left( {\matrix{ 0 & 1 & 2 \cr 0 & 0 & 0 \cr } } \right)$$ I want to find a basis for the row-space of the matrix above. One might say $$B = \left\{ {\left( ...
0
votes
2answers
279 views

How do I show this

Given invertible matrices $A,B$ and $P$ such that $A = PB$, then we say that $A$ is left equivalent to $B$. Show that left equivalence is indeed an equivalence relation.
1
vote
2answers
68 views

Why is it true that for every eigen value a: rank $(T^*- a^*I) =$ rank $(T-aI)$?

I've seen a lemma: rank$(T^*- a^*I) =$ rank$(T-aI)$ where $T^*$ is $T$ adjoint and $a^*$ is the adjoint eigen value. I know that $T$ and $T^*$ has the same rank, but can someone help me understand ...
3
votes
0answers
69 views

Continuous subgroup of SO(3)?

I read from a paperarXiv: cond-mat/0602109 by a theoretical physicist, Prof. Frank Bais, close subgroups of $SO(3)$ is given by ${C_n,D_n,T,O,I,SO(2)\rtimes Z_2}$, where $C_n$ is the cyclic group of ...
0
votes
1answer
57 views

Confusion over Matrix rotation

I want to make a function in C++ that accepts an angle 'a', and a vector 'v' as arguments and returns a matrix. 'a' should represent the amount that is rotated around vector 'v', an arbitrary axis, ...
0
votes
1answer
57 views

Eigenvalues and eigenvectors general $n \times n$

Find the eigenvalues and eigenvectors for the general $n \times n$ matrix which has $2$'s across the main diagonal, $-1$'s below and above the main diagonal?
2
votes
1answer
441 views

Notation for the set of symmetric matrices and symmetric positive definite matrices

I would like to know if there exists a notation for the set of symmetric matrices and symmetric positive definite matrices. For instance, the set of $N \times N$ matrices with real entries is denoted ...
0
votes
1answer
33 views

Proof regarding effect of row operations on determinants>

Let $A,B \in K^{n,n}$ and suppose $B$ is obtained from $A$ by adding $\lambda$ times row $j$ to row $i$. Prove $det(A)=det(B)$. My Attempt I tried to use proof by induction for this . Take ...
0
votes
0answers
48 views

Simultaeous diagonalizability and the commutator

Is there any intuitive, down-to-earth, reason why the vanishing of the commutator $[ \ , \ ]$ of two operators $A$ and $B$, $[A,B] = 0$, implies simultaneous diagonalizability of $A$ and $B$, and why ...
1
vote
1answer
43 views

Subgroups of GL(2,C) isomorphic to Z

Let $\mathbb Z\to \mathrm{GL}_2(\mathbb C)$ be an injective homomorphism. I'm wondering about the possibilities for the image of $\mathbb Z$. I think the image is always conjugate to a subgroup of ...
0
votes
1answer
44 views

Solve Matrix: How many trips can be made?

We have the matrix: $$\mathbf{M}=\begin{bmatrix}0 & 1 & 1 & 0 \\ 1 & 0 & 1 & 0 \\ 1 & 0 & 0 & 1 \\ 0 & 1 & 1 & 0\end{bmatrix}$$ The matrix ...
1
vote
0answers
112 views

Meaning of $A^A$

Let $A$ be a matrix. I know that there is definition for $A^k$ power of $A$, and $e^A$ expontential of $A$. Is there any meaning of $A^A$?
0
votes
2answers
51 views

Rotation counterclockwise

Let $A_{\theta}$ be rotation counterclockwise by $\theta$ as follows: $$A_\theta = \left[ \begin{matrix} \cos(\theta) & -\sin(\theta)\\ \sin(\theta) & \cos(\theta)\end{matrix} \right]$$ ...
0
votes
2answers
331 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.
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0answers
52 views

Number of combinations in a matrix

Given the size of a matrix is $N \times N$, how many unique matrices are there given the following restrictions: Matrix entries can only contain numbers $\left[0,b\right]$ A valid matrix cannot have ...
1
vote
1answer
48 views

linear transformations with matrices $A, A^*$

Let $K$ be a field, $K\subseteq \Bbb C$. $V$ is a linear space over $K$, $\dim(V)=n(n\geq2)$. Choose ordered basis $\epsilon_1,\epsilon_2,\dotsc,\epsilon_n$ for $V$. $\bf A,B$ are two linear ...
1
vote
1answer
82 views

Orthogonal projection of a vector $x$ onto $\operatorname{Col}(A)$

Let $A$ be a matrix. The orthogonal projection of a vector $x$ onto $\operatorname{Col}(A)$ is unique if and only if the columns of $A$ are linearly independent. True or False?
4
votes
2answers
79 views

Compute $ \lim_{n\to \infty}\prod_{i=1}^n B(p_i^{-2})$

Let $B(x) = \begin{pmatrix} 1 & x \\x & 1 \end{pmatrix}$, and $2=p_1<p_2<\cdots <p_n <\cdots$ primes number. Compute $$\displaystyle \lim_{n\to \infty}\prod_{i=1}^n ...
0
votes
0answers
29 views

product of Mueller matrices

I am reading very old Fortran77 program. In the program, there is a subroutine for the product of two 4 x 4 Mueller Matrices. The subroutine produces matrix "C" using matrix "A" and matrix "B". The ...
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vote
0answers
37 views

spectrum of a special class of tridiagonal matrices

Consider a real and symmetric tridiagonal matrix with zero diagonals and where subdiagonals and superdiagonals are equal to 1 except the (1,2)-th component being equal to $a$, i.e., $\begin{bmatrix}0 ...
0
votes
1answer
16 views

A matrix to form a shorter vector from a given vector by discarding some elements of the latter.

I will try to explain by an example: Given: A 16x1 vector $ V = [a, b, c, d, e, f, g, h, i, j, k, l, m, n, o, p,]^T$ and four 9x1 vectors $R_1,R_2,R_3$ and $R_4$. Required: Four 9x16 extraction ...