1
vote
1answer
20 views

Find the $n^{th}$ power of a $2$x$2$ matrix.

Let $A=\begin{pmatrix}3&-2\\2&-2\end{pmatrix}$. Using Lagrange's interpolation compute $A^n$ for $n\in\mathbb{N} $ So far I've worked out the minimum polynomial of $A$ to be $(x-2)(x+1)$ but ...
2
votes
1answer
33 views

Moving a point around a circle

we're currently working on a game which involves a character that rotates around a point. We are using a rotation matrix to rotate a given a point (x,y) around another point by first translating to ...
4
votes
4answers
91 views

Why is the volume of a parallelepiped equal to the square root of $\sqrt{det(AA^T)}$

Why is the $\sqrt{det(AA^T)}$ equal to the volume of a parallelepiped? Is is somehow related to the fact that $det(A) = det(A^T)$? EDIT: To clarify, the parallelepiped is spanned by the columns of ...
0
votes
1answer
22 views

Norm of the sum of inverse matrices

Let $A,B$ be two invertible matrices. Is there a way to compute $\|A^{-1} -B^{-1}\|$ in terms of $\|A-B\|$?
3
votes
0answers
42 views

Proving that table totals can always be preserved with ceiling and floor

$\begin{array}{|c|c|c|c|} \hline 11.998& 9.083 & 2.919 & &24 \\ \hline 12.983&10.872&3.145&&27\\ \hline 1.019&2.045&0.936&&4\\ \hline & & ...
1
vote
1answer
28 views

Is $vv^{T} - v^{T}vI$ non-singular? [on hold]

Is $vv^{T} - v^{T}vI$ non-singular ? Why? $v$ is vector
0
votes
1answer
8 views

Number of Distinct Elements in Set of Products of 2 Matrices

Let $X=\begin{pmatrix}\cos\frac{2\pi}{5} & -\sin\frac{2\pi}{5}\\\sin\frac{2\pi}{5} & \cos\frac{2\pi}{5}\end{pmatrix}$ and $Y=\begin{pmatrix}1 & 0\\0 & -1\end{pmatrix}$. Find the ...
1
vote
0answers
26 views

Eigenvalues with constraints?

Note: This is a short version of About diagonalizing a matrix for a quadratic expression (with the goal of uncoupling mixed terms) For a $n$-dimensional symmetric matrix A, orthogonal matrix C exists ...
0
votes
1answer
21 views

Linear Algebra - Give an example for $3x3$ matrix for these eigenvalues

I'm having trouble with this problem : Give an example for matrix $A$ with these eigenvalues $\lambda_1-1,\lambda_2=1,\lambda_3=0$ while : $$v_1=(0,1,1)$$ $$v_2=(1,-1,1)$$ $$v_3=(0,1,-1)$$ ...
4
votes
4answers
177 views

Question about determinants

I am working on some practice problems and I just cant see to even begin to understand how to do this question. It starts off by giving some facts such as det= 1 for the following:$$ \begin{matrix} a ...
0
votes
1answer
19 views

Need to prove $(JC=0=CJ,\,JJ=nJ)\implies (C-aJ)^{-1}-(C-bJ)^{-1}=\frac{b-a}{ab n^2} J$

I can't prove that matrix $C$: $$\big(JC = 0 = CJ\text{ and } JJ = nJ\big) \implies \left((C-aJ)^{-1} - (C-bJ)^{-1} = \frac{b-a}{abn^2} J\right)$$ I know that $$(JC = 0 = CJ\text{ and }JJ = nJ) ...
2
votes
1answer
26 views

Get normalised eigenvectors

I am given the matrix: $\begin{pmatrix} a & b \\ b & -a \end{pmatrix}$ and I already calculated the eigenvalues $\lambda = \pm \sqrt{a^2+b^2}$. Now, I want to get the normalised ...
0
votes
1answer
25 views

Element matrix multiplication representation

Matrix element by element multiplication defined : $C=A*B$ $c_{ij}=a_{ij}b_{ij}$ Is this multiplication can be represented with stardant matrix multiplication or Kronecker product ?
2
votes
3answers
60 views

Roots of a cubic equation with coefficients based on unknown values $a$, $b$ and $c$.

I want to find the eigenvalues of the following matrix: $$ \left( \begin{array}{ccc} 0 & a & b \\ a & 0 & c \\ b & c & 0 \end{array} \right) $$ So, I found the characteristic ...
0
votes
0answers
20 views

Proof involving projections and column spaces

Let $A \in \mathbb{M}_{m×n}(\mathbb{R})$ with linearly independent columns. If $\overrightarrow{b} \in \mathbb{R}^m$, then prove $proj_{Col(A)}(\overrightarrow{b}) = ...
1
vote
3answers
46 views

Prove that a matrix with a given characteristic polynomial is diagonalizable

Matrix $A$ is defined over real number. Characteristic polynomial : $p(x)=(x+3)^2(x-1)(x-5)$ It also known that : $$\text{rank}(A+2I)+\text{rank}(A+3I)+\text{rank}(A-5I)=9$$ prove $A$ ...
0
votes
1answer
28 views

Domain/codmain + range/kernel for linear mappings

Consider the linear mapping: $$L(x_1,x_2)=(2x_1-3x_2,4x_1+5x_2,2x_1-x_2)$$ Solve for: (a) Domain and codomain of L (b) Standard matrix of L (c) Basis for the range of L (d) Basis for the kernel ...
0
votes
1answer
19 views

Is the derivative of the characteristic polynomial equal to the sum of characteristic polynomial of principle submatrices?

Let $A$ by an $n \times n$ matrix over the complex numbers and let $\phi(A,x) = \det(xI-A)$ be the characteristic polynomial of $A$. Let $B_i$ be the principal submatrix of $A$ formed by deleting the ...
1
vote
2answers
40 views

Show that a set of vectors is linearly dependent

Show that the set $S = \{(3, 2), (−1, 1), (4, 0)\}$ is linearly dependent by finding a nontrivial linear combination of vectors in the set whose sum is the zero vector. (Use $s_1$, $s_2$, and ...
-1
votes
1answer
29 views

Linear Algebra - Prove trival solution eigenvalue

A is an $2\times2$ matrix with $\operatorname{trace}=1$, and $\det A=-6$. Prove that $(2A+5I)x=0$ has only trival solution. I need to show that $(-A-\frac{5}{2}I)x=0$ Therefore I need to show that ...
0
votes
1answer
29 views

Can't understand matrix based derivation

$\beta(k,d)=(X'X+kI)^{-1}(X'y+kdB_L)$ $=[I+k(X'X)^{-1}]^{-1}(X'X)^{-1}(X'y+kdB_L)$ $=[I+k(X'X)^{-1}]^{-1}(B_L+kd(X'X)^{-1}B_L)$ $=[I+k(X'X)^{-1}]^{-1}(B_L- dB_L)+dB_L$ $B_L=(X'X)^{-1}Xy$ X=n*p ...
6
votes
1answer
50 views

What is the quickest way to find the characteristic polynomial of this matrix?

Let $e_k$ be the $k$-th vector of the canonical base of $\mathbb R^n$ and let $$B = [e_2 \mid e_3 \mid \dots \mid e_n \mid e_1]$$ What it the quickest way to show that the charachteristic polynomial ...
1
vote
1answer
34 views

Gram matrix to be cancelled

Let $V$ be a $n$ dimensional Euclidean space with inner product $<\cdot,\cdot>$, with basis $e_1,\cdots,e_n$. Then the Gram matrix is $A=(a_{ij})$ with $a_{ij}=<e_i,e_j>$. It is well-known ...
1
vote
2answers
28 views

Positive definite matrix to be cancalled

From $ax\geq 0$ for $a>0$, we have $x\geq 0$. So I suggest that if $Ax\geq 0$ for $A$ positive definite matrix, $x$ a column vector, $0$ is the column vector with $0$ as elements, then $x\geq 0$, ...
4
votes
1answer
43 views

Finding unknown matrices in a set of simultaneous matrix equations

I've come across a thorny problem in my research, which is too complicated and specific to ask here. However, it bears some similarity to the following problem, and understanding how to solve this ...
3
votes
0answers
47 views
+50

Proof of the conjecture that the kernel is of dimension 2, extended

Pursuing my research, I am now looking for a proof of an extension of the problem proposed here and answered. It's an extension in the sense that I'm now considering two different $t_1$ and $t_2$. The ...
0
votes
1answer
38 views

Linear Algebra vs Matrix Algebra [on hold]

Hi I don't know if this would be a proper question to ask here or not Anyway, I am an undergraduate electrical engineering student and I am considering taking another math course. What is the ...
0
votes
1answer
30 views

On Neumann-series of matrices

Let $A \in \mathbb{R}^{n \times n}$ and we denote with $\rho(A)$ the spectral radius of $A$ and with $I_n \in \mathbb{R}^{n \times n}$ the identitiy matrix. Applying Carl Neumann's result on matrices ...
2
votes
1answer
47 views

about the power of a matrix

Assume that matrix $A$ contains only 0 or 1 elements. Could anyone give me some condition, under which the matrices $A^i$ (for $i=1,2,3,...,k$) still contains only 0 or 1 elements. For example, I ...
0
votes
0answers
31 views

Two matrices with the same row space are row equivalent

If $\operatorname{Row} A = \operatorname{Row} B$, then $A $ is row equivalent to $B$. So far I have So $\operatorname{Row}(B) = \operatorname{Row}(A)$. That is rows of $B$ belong to ...
0
votes
1answer
35 views

Find the standard matrix of the linear transformation

Suppose there is a linear transformation $T:\mathbb{R^2} \rightarrow \mathbb{R^2}$ such that $$T\left( \begin{array}{ccc}2 \\ 1 \end{array} \right)=\left( \begin{array}{ccc}1\\ 4 \end{array} ...
0
votes
0answers
14 views

Reduce to diagonal form.

Problem is to reduce $5X^2+3Z^2+4XY-4YZ+6ZX$ into diagonal form over $\mathbb{R}$. With my knowledge, We ned to make a non-singular variable transformation so that above form comes into a form like ...
-1
votes
1answer
25 views

Diagonalizing a block matrix

Suppose the matrices $A,B,C,D$ are $M\times M$, $M\times N$, $N\times M$ and $N\times N$, respectively. Then I give you the following block matrix: $$ G = \begin{pmatrix} A & B \\ C & D ...
1
vote
2answers
77 views

Linear Algebra - Prove $AB=BA$

Let $A$ and $B$ be any $n \times n$ defined over the real numbers. Assume that $A^2+AB+2I=0$. Prove $AB=BA$ My solution (Not full) I didn't managed to get so far. $A(A+B)=-2I$ ...
0
votes
1answer
21 views

Show that $P(A) = 0\implies P(B) = 0$ if matrix $A$ is similar to matrix $B$ and $P(x)$ is any polynomial.

Note: This is a homework problem. I've been working on this one for a while now and seem to be a bit stuck. The one thing for sure I know we can say is that $P(A) = P(M^{-1}BM) = 0$ Is there some ...
2
votes
3answers
24 views

Solving for a matrix in equation form

Solve for $X$ assuming all matrices are n x n and invertible as needed. $$B(X+A)^{-1}=C$$ I solved this the following way: Multiply both sides by $(X+A)$ Multiply both sides by the inverse of $C$ ...
0
votes
0answers
19 views

Linear transformation on finite-dimensional vector space such that T²=T, how is the matrix?

I'm trying to have a picture of how the matrix of such a linear transformation looks like and why? I can't really find anything but that applying the map once can change the input into a new output ...
0
votes
2answers
27 views

Solving $CT = PC$ for transforms in $SE(3)$

I have three transforms: $C$, $T$, and $P$. Each of these transforms consists of 3D rotations and translations. I know $T$ and $P$, and I would like to solve for $C$. They are related by $T = C^{-1} P ...
1
vote
2answers
47 views

The inverse of AR structure correlation matrix / Kac-Murdock-Szeg ̈o matrix

I want to find the inverse of the following matrix: $$ R_{k-1}=\begin{pmatrix} 1 &\rho &\rho^2 &\cdots &\rho^{k-2} \\ \rho &1 &\rho &\cdots ...
3
votes
1answer
28 views

Is this argument on positive definite matrices correct?

Let $A$ be a $N\times N$ positive definite matrix. Then, there exists a $N\times 1$ gaussian random vector $a$ such that $A=E[aa^T]$ where $E[.]$ denotes expectation. Then for any given vector $x$, ...
1
vote
0answers
28 views

Fast Cholesky Factrorization for Tree Laplacians

Suppose $T_1$ and $T_2$ represent two Laplacian matrices of two spanning trees of $n$ vertices. Since the Cholesky factorization needs $O(n)$ time for each $T_i\ (i=1,2)$ due to the tree structure, ...
1
vote
1answer
50 views

Proof that if $A$ is similar to $B$, then $B$ is similar to $A$

$A$ is similar to $B$ if there is an invertible matrix $S$ such that $B = S^{-1}AS$. Prove that if $A$ is similar to $B$, then $B$ is similar to $A$. So if $A$ is similar to $B$ then $B = ...
2
votes
3answers
60 views

What is inverse of $I+A$ given that $A^2=2\mathbb{I}$?

I have the next problem: Let $A$ be a real square matrix such that $A ^ 2 = 2\mathbb{I}$. Prove that $A +\mathbb{I}$ is an invertible matrix and find its inverse. I tried with the answers given ...
0
votes
2answers
40 views

Does $TS$ being an isomorphism imply that $S$ is an isomorphism?

Let $V, W, U$ be vector spaces over a field $\Bbb F$, and suppose that $S : V → W$ and $T : W → U$ are linear. If $TS$ is an isomorphism, is it true that $S$ must be an isomorphism? If it's not ...
1
vote
0answers
33 views

Linear Algebra - Find basis for $ImT$ and $KerT$.

$B=(u1,u2,u3) \in R^3$ $u1=(1,-1,0)$ $u2=(1,1,1)$ $u3=(1,2,3)$ $T : R^3 \rightarrow R^3$ This is expression matrix (not sure if that the right term in English) on basis B: $[T]_B ...
0
votes
1answer
23 views

Eigen values of a positive semidefinite matrix and its transpose

$A\in M_n(\mathbb{C})$ is positive semi-definite so there there exists unitary matrix $U$ such that $A=U^*DU$ where $D$ is the real diagonal matrix consisting of eigen values $(\ge 0)$ of $A$, now I ...
0
votes
1answer
43 views

Expressing determinant as a linear combination of minors of fixed dimension

Suppose $k<n$. How does one express $\det\begin{pmatrix}a_1^1&\dots&a_n^1\\ \vdots&\ddots&\vdots\\ a^n_1&\dots&a^n_n\end{pmatrix}$ in terms of a linear combination of ...
-1
votes
0answers
36 views

Linear Alegbra - The $U + W$, $U \cap W$, $U \cup W$, $U-W$ of subspaces and not subspaces

Lets assume $U,W \subseteq \mathbb{R}^4$ $U=\{u_1 = (0,0,0,1), u_2=(1,0,0,0)\}$ $W=\{w_1=(0,0,1,0),w_2=(1,0,0,0),w_3=(0,1,0,0)\}$ I understand that in case $U$ and $W$ are not subspaces: Case $U ...
8
votes
3answers
484 views

Proving or disproving A+B is invertible

Given two matrices $A,B\in M_n (F)$, where $A$ is $k$ -nilpotent and $B$ is invertible, is it true that $A+B$ is also invertible? I was having trouble on how to prove this, and then I thought maybe ...
0
votes
1answer
56 views

Linearly dependent eigenvectors of a matrix

I read a theorem that says squared matrix $A_{n\times n}$ is diagonalizable iff there is a set of $n$ linearly independent vectors ,each of which is an eigen-vector of $A_{n\times n}$ . I understand ...