Questions related to equations, with matrices as coefficients and unknowns.

learn more… | top users | synonyms

0
votes
2answers
20 views

Finding a matrix inverse when an equation involving it is a multiple of the identity matrix

Say you had a matrix $A$, and you did an equation like $A^2 - A$, and proved that it was a multiple of $I$. How could you find $A^{-1}$ in the form $rA + sI$ after proving that? I want to do it ...
1
vote
0answers
48 views

Realation between Matrices ?!

The answer to the following question could be trivial. Let $A_1, A_2$ be symmetric $n\times n$ matrices, $x=(x_1,\ldots,x_n)\in \mathbb{R}^n$. If the maximum is taken for over ($\|x\|=1,\, and ...
0
votes
1answer
54 views

Wolfram|Alpha refuses to find the inverse of a large 6x6 matrix.

Just to be clear, this isn't a question on how to find the inverse of a matrix, I just don't want to find the inverse by hand (I hope you see why). $$ \begin{pmatrix} 1 & 2006 ...
0
votes
0answers
25 views

$A$ is nonsingular. eigenvalues of $A$ are unequal to zero

Assume that $A$ is nonsingular. Show that all eigenvalues of $A$ are unequal to zero. Express the eigenvalues and eigenvectors of $A􀀀^{-1}$ in terms of those of $A$.
0
votes
0answers
31 views

In matrix algebra, what's the name for the inverse operation of pre- or post- multiplication?

For example, in this typical equation: $$\mathbf{Mv}-\lambda \mathbf{v}=\mathbf{0}$$ (where $\mathbf{M}$ is a symmetric matrix, $\mathbf{v}$ is a vector, $\lambda$ is a scalar, and $\mathbf{0}$ is a ...
0
votes
1answer
27 views

How can a non-zero matrix $A$ be found such that Adj$(A) = 0$? [on hold]

Is it possible to find a $3 \times 3$ matrix $A$ such that it's adjoint is $0$?
0
votes
0answers
22 views

Finding transform matrix from resulting multiplypoint function

Two matrix transformation functions exist within the Unity3D API: 1) MultiplyPoint 2)MultiplyPoint3X4 3X4 matrix (2) preforms a standard transform against a vector (And ofc is easily replicated ...
1
vote
2answers
28 views

Matrix problem invovling orthogonal matrices

If $$P=\begin{bmatrix} \cos\theta & \sin\theta \\ -\sin\theta & \cos\theta \end{bmatrix}\text{ and } A=\begin{bmatrix}1 & 1\\0 & 1\end{bmatrix}$$ And $Q=PAP'$ then ...
-2
votes
1answer
20 views

Problem with prove about matrix [closed]

Prove if $A,B$ invertible and then $(A^{-1}B)^t$ is invertible Thanks
0
votes
0answers
23 views

how to solve this differential equation in matrix form?

I am reading the article [1] and came across a differential function that the authors did not provide detailed steps of the solution. The function is of great interest to me. Any thoughts on how the ...
1
vote
0answers
39 views

Geting $A$ from $AA^{T}$ [duplicate]

I have a symmetric matrix $B$ (actually a covariance matrix of a set of variables) and I want to write it in the form of $AA^{T}$. How can I get $A$? Thanks.
1
vote
0answers
17 views

What are the different ways of performing Triangular matrix-vector multiplication?

Suppose we have $$\left[\begin{array}{cccc} x_1 & 0 & 0 & 0 \\ x_2 & x_1 & 0 & 0 \\ x_3 & x_2 & x_1 & 0 \\ x_4 & x_3 & x_2 & x_1 \end{array}\right] ...
0
votes
0answers
40 views

A generalized eigenvalue problem

The generalized eigenvalue problem likes this: $\begin{pmatrix} 0 & C_{12}\\ C_{21} & 0 \end{pmatrix} \begin{pmatrix} \xi_1\\ \xi_2 \end{pmatrix}=\rho\begin{pmatrix} C_{11} & 0\\ 0 ...
0
votes
0answers
18 views

Finding Inverse of a matrix using elementary transformations

So I have to find the Inverse of A. $$ A = \begin{bmatrix} 1 & 2 & 3 \\ 1 & 3 & 4 \\ 3 & 4 & 3 \\ \end{bmatrix} $$ By using elementary row or column transformations.. The ...
-1
votes
1answer
28 views

Solving linear equations in Matlab

I am trying to solve the problem $Ac = b$ but I am having difficulty figuring out how to set it up. I thought multiplying by ${\rm e}$ would remove the $\ln$ but I guess I forgot my basic math. My ...
3
votes
3answers
70 views

Say for what values of $a \in \mathbb {R} $ this matrix system has solutions

Let $a \in \mathbb{R}$ and $$ A_a = \begin{pmatrix} 1 & a & 1 \\ a & 2 & 3 \\ 2 & 3 & 4 \end{pmatrix}. $$ Say for each values of $a \in \mathbb{R}$ the ...
2
votes
1answer
67 views

Prove that $\int_{0}^\infty (e^{As})^T e^{As}\,ds$ is convergent

Consider A to be a matrix that has all eigenvalues $\lambda$ with negative real part, that is, $Re(\lambda) < 0$. a)Show that the integral $\int_{0}^\infty (e^{As})^T e^{As}\,ds$ is convergent, ...
1
vote
1answer
31 views

How to neatly summarize indexes of a matrix where there are a lot of i's x j's [closed]

As you can see from the subject line, I can't even think of the word of what I need to do. I am trying to write in text that I multiplied columns of a matrix (n columns, i = 1:n). There are many ...
0
votes
1answer
39 views

Derivation wrt a vector variable: what happens to transpose of the vector?

Considering $x,y$ are vectors and $\mu,\Sigma$ are mean (vector) and covariance (matrix); how to solve: $(1): \displaystyle \frac{\partial }{\partial X}{[(y-x-\mu_N)^T\Sigma_N^{-1}(y-x-\mu_N) + ...
0
votes
1answer
24 views

Matrix Algebra: finding what values of a does system have nontrivial solutions

Instructions are to: find what values of 'a' does system have nontrivial solutions The original equations are \begin{equation} x+2y+z=0 \end{equation} \begin{equation} -x-y+z=0 ...
0
votes
0answers
16 views

Convergence of Adjacence Matrices

I have a graph, with some nodes, and weighed edges with positive weights only. The sum of the weights from one node is less or equal to 1. There will be at least one node, that the sum is less then 1. ...
0
votes
2answers
50 views

Defining Unitary Matrices

I have got a question and I would appreciate if one could help. I start with an example to explain what I am looking for. Assume a scaled unitary matrix like $U_2 = \begin{bmatrix} 1 & 1 \\ 1 ...
0
votes
0answers
21 views

trace inequality trivial on kernel

Let $x_1, x_2>0$ such that $x_1+x_2=0$. Then the concativity theorem claims that for any $n \times n$ matrix $ K$ and any positive matrices $A_1, A_2$ following inequality holds for all ...
0
votes
0answers
40 views

Write the Following in matrix notation

Write the following in matrix notation: $$\matrix{i' &=& \cos(wt)i - \sin(wt)j\\ j' &=& \sin(wt)i + \cos(wt)j\\ k' &=& k}$$ Note: $i, j$ and $k$ are all vectors! Show ...
0
votes
1answer
13 views

Singular value decomposition of 2x2 matrix with unit norm entries

I've got a question and I would appreciate if one could help me to understanding it. I have a 2x2 complex matrix $F$. The absolute value of the entries of $F$ are equal to one, i.e., $|F(m,n)|=1$. I ...
1
vote
1answer
46 views

Can I think of this as an eigenvalue/eigenvector problem?

PREMISE: Suppose you are given a matrix $S$ and you have to find a projector $P$ (which is a hermitian matrix that satisfies $P^2=P$) that is a solution of $$PSP=\lambda(S)P,$$ for $\lambda(S)$ some ...
1
vote
0answers
16 views

Requirements for $\mathbf Q = \mathbf P\cdot\mathbf Q^T\cdot\mathbf P^T$

Say I have a known matrix $\mathbf Q$, outer product of 2 vectors. What kind of matrix $\mathbf P$ do I need for the following statement to be true? $$\mathbf Q = \mathbf P\cdot\mathbf ...
0
votes
1answer
26 views

Product of matrices: $\mathbf Q^{-1}\cdot\mathbf B^T\cdot\mathbf B\cdot\mathbf Q$

Let $\mathbf Q$ be an orthogonal matrix ($\mathbf Q^T\cdot\mathbf Q=\mathbf I$). If I have another matrix $\mathbf B$, is there any special propriety found in the following expression? $$ \mathbf ...
0
votes
0answers
22 views

the solution of a system of nonlinear differential equations [duplicate]

Consider a system of differential equations can be written on the form $$ i\frac{\partial}{\partial t}\mathcal{A}(t)=\mu(t) \mathcal{A}(t) $$ where $\mu(t)$ is the coefficient matrix and is time ...
0
votes
1answer
26 views

Isolate vector in matrix equation

I have this equation: $$H=\frac{t\cdot n^T}{n^T\cdot x}$$ with $t$, $n$ and $x$ being $3\times 1$ column vectors and $H$ a $3\times 3$ matrix, and where $\cdot$ is matrix multiplication. Notice that ...
2
votes
1answer
46 views

Find solution to matrix equation

Find all solutions to the equation $X^2+ \begin{bmatrix} 1 & -1 \\ 1 & 1 \\ \end{bmatrix}X+\begin{bmatrix} -7 & 1 \\ 0 & 0 \\ ...
1
vote
1answer
23 views

How is it the matrix-vector multiplication with SVD is $O(m+n)$?

Assuming the singular value decomposition is known, how is it the matrix-vector product of $\mathbf{A}$ ($m \times n$) and vector $\mathbf{x}$ ($n \times 1$) has $O(m+n)$ complexity? Somewhat related: ...
4
votes
0answers
48 views

Determinant of a function

I was thinking about matrices and then why arent there matrices with uncountable many values? (Probably this conecpt already exists for a very long time, but i don't know it) Assume there are ...
-1
votes
1answer
39 views

Solving matrix using Gaussian elimination and a parameter

$\begin{bmatrix} x_{1} & 2x_{2} & & & ax_{5} & x_{6} & = & -2 \\ -x_{1} & -2x_{2} & & & (-1-a)x_{5} ...
1
vote
1answer
33 views

Matrix notion for a double summation

My question is some-what tied to this one: Is there any matrix notation the summation of a vector? Is there a way to express this double summation into an equivalent product of matrices/vectors? ...
1
vote
1answer
24 views

Minimal polynomial of a $4\times4$ matrix [closed]

I just need to see an example of a non-diagonalizable $4\times4$ matrix over $\mathbb{R}$ whose minimal polynomial is the same as its characteristic polynomial. I saw the question elsewhere and ...
0
votes
0answers
31 views

Diagonalize and square matrix

Diagonalize $\begin{bmatrix} 5 & 0 & 0\\ 1 & 5 & 0\\ 0 & 1 & 5\\ \end{bmatrix}$ and find P and D such that $P^{-1}AP=D$ So I created the matrix $\begin{bmatrix} \lambda-5 ...
1
vote
2answers
31 views

Find a basis for the orthogonal complement of a matrix

Let $S \subset \mathbb{R}^4$ be the vectors whose components satisy $x_1 + x_2 - x_3 + x_4 = 0$ Find the dimension of S and then find a basis for the orthogonal complement of S So to find ...
9
votes
3answers
988 views

Is this possible? AB- BA=I [duplicate]

I have just started linear functionals when I faced the following problem: If $A$ and $B$ are $n \times n$ complex matrices, show $AB - BA=\Bbb{I}$ is impossible. Can someone help me?
0
votes
0answers
118 views

Question about an equation using linear algebra

Firstly, in $n \in \bf{N}$, we consider the square matrices as $\bf{\{H_{i},A_{i},P,Q \}} \in \bf{R}^{\it{n \times n}}$ and the vectors $\bf{\{ a,x,1 \}} \in \bf{R}^{\it{n}}$. All parameters are known ...
0
votes
0answers
17 views

Rewriting residuals transposed times estimates with Annihilator and Projection: $e'\hat{y} +0 = y'M_xP_xy =0$

I am looking at a proof that says $$e'\hat{y} +0 = y'M_xP_xy =0$$ where $P_x$ is the projection matrix, $M_x$ is annihilator. I don't see how they obtain this result. What I am wondering is they are ...
0
votes
3answers
28 views

Find a matrix from its eigenvalues and corresponding vectors

Suppose $A$ is a $3 \times 3$ matrix with eigenvalues $\lambda_1=-1$ $\lambda_2=0$ and $\lambda_3=1$ and with the corresponding eigenvectors $\vec{v_1}=<1,0,2>$ $\vec{v_2}=<-1,1,0>$ ...
3
votes
2answers
89 views

Find the $30^{th}$ power of a matrix

Let $A$ be the following matrix : $$A=\left(\begin{array}{rrr} 1&0&0\\ 1&0&1\\ 0&1&0\\ \end{array}\right) $$ Find the matrix $A^{30}$. Is this matrix diagonalizable? Is there ...
1
vote
2answers
65 views

Find the nth power of a matrix

Let matrix A is $$A=\left (\begin{array}{rrr} 1&0&0 \\ 1&1&0\\ 0&0&1 \end{array}\right)$$ How can I find the 30 th power of A.. Is diagonalization possible? I found the ...
1
vote
1answer
27 views

finding $K$ with $KQK^∗=0$ and $KK^∗=I$

For a given nonsingular complex matrix $Q_{m\times m}$, how can we find a complex $K_{n\times m}$, $n<m$, with \begin{align*} KQK^∗&=0\\ KK^∗&=I? \end{align*}
1
vote
1answer
49 views

How to extend the parallelepiped volume formula to higher dimensions?

The volume of a parallelepiped $(V)$ is given by the triple scalar product: $$V=\mathbf{c}\cdot{}(\mathbf{a}\times\mathbf{b})$$ where $\mathbf{a}$, $\mathbf{b}$, and $\mathbf{c}$ are the vectors ...
1
vote
3answers
29 views

Why does it seem that eigendecomposition requires that the decomposed matrix be diagonal?

The eigen-decomposition of positive semi-definite matrices always exists. Given such a matrix $\mathbf{A}$, then, we have $$\mathbf{Av}=\lambda\mathbf{v}$$ for a given eigen value $\lambda$ and ...
1
vote
2answers
114 views

Why using determinant equal to zero

I am stuck at a step in some calculations from my teacher: we have that $x_1$ and $x_2$ are non-zero and that $u_1>0$. $ \begin{bmatrix} u_1 & 4 \\ 4 & u_1 \\ ...
0
votes
4answers
74 views

How do I know the span of a matrix

I have a marix that is $$R=\pmatrix{1&1&1\\1&1&1+2a\\0&2&2+2b}$$ and the span of R is span=$\left\{\pmatrix{1\\1\\0},\pmatrix{0\\0\\1}\right\}$ In my preliminery thinking, ...
0
votes
2answers
95 views

How to solve for $A$ in $A - BAB^T = CC^T$?

Considering an unknown real symmetric matrix $A$, and two known matrix $B$ and $C$. If we have the equation: $$ A - BAB^T = CC^T $$ Can we get an analytical solution of A?