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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91 views

Positive Definite Matrix Question

I am having some little difficulties to show the following: Let $M$ be a symmetric positive square matrix. I must show that $I-M$ is positive definite matrix if and only if $M^{-1}-I$ is positive ...
1
vote
2answers
466 views

Euclidean norm and matrix questions

Let $A$ be an $n×m$ real matrix and define $\vert A \vert_{2}^{2}={\rm tr}(A^tA)$. 1)Show that $|A|_2$ is the Euclidean norm of $A$, when we view $A$ as a vector in $R^{nm}$ by stacking the columns ...
0
votes
3answers
249 views

Determine the value of A & B for this system ?!!!

Determine the value of A & B for which system $$ \begin{pmatrix} 3 & -2 & 1 \\ 5 & -8 & 9 \\ 2 & 1 & A \\ \end{pmatrix} \begin{pmatrix} x \\ y \\ ...
2
votes
2answers
122 views

Minimal polynomial and Cayley hamiliton theorem

My Classmate asked me two following question this morning. But I still can't figure out proper solutions... (1) If $A \in R^{nxn}$, satisfy $A^{3}+A+I_n=0$, find$a,b,c\in R$ such ...
2
votes
0answers
584 views

How to calculate the submatrix inverse with prior knowledge of matrix inverse?

Given $A\in \mathbb{N}^{n\times n}$, then $A(\mathcal{I})$ is defined by first deleting the those columns with index in $\mathcal{I}$ and then extracting the first $n-|\mathcal{I}|$ rows. Note that ...
0
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1answer
46 views

Please calculate the formula for $\varphi$

Define $$\varphi:\mathbb{R}^{3}\rightarrow\mathbb{R}^{3}$$ $$A:((3,1,1),(1,0,0),(5,1,0))$$ $$B:((3,4,5),(4,1,1),(2,0,1))$$ Consider $\varphi$ as linear transformation with matrix ...
7
votes
3answers
362 views

Solving matrix equation $XA=AY$ with known $X$ and $Y$

I am having problem in solving set of matrices multiplication. There are three matrices $A,X$ and $Y$, all are non-singular $2\times 2$ matrices. Where matrix $X$ and $Y$ are known and $A$ is unknown. ...
1
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0answers
75 views

transformation of symplectic structure by a matrix

Suppose that in canonical symplectic basis $e_1,e_2,f_1,f_2$ we have $$\Omega=pf_1^*\wedge f_2^*+qe_1^*\wedge e_2^*+r(e_1^*\wedge f_2^*+e_2^*\wedge f_1^*)+s(e_1^*\wedge f_1^*-e_2^*\wedge f_2^*)$$ Let ...
2
votes
0answers
75 views

Kalman filter implementation question

I have the following code to define a Kalman filter: ...
2
votes
1answer
56 views

$\int_{\mathbb{R}^n} dx_1 \dots dx_n \exp(−\frac{1}{2}\sum_{i,j=1}^{n}x_iA_{ij}x_j)$?

Let $A$ be a symmetric positive-definite $n\times n$ matrix and $b_i$ be some real numbers How can one evaluate the following integrals? $\int_{\mathbb{R}^n} dx_1 \dots dx_n ...
2
votes
1answer
115 views

Lagrange identity on matrices

Consider the $k-$dimensional homogeneous linear difference system $ x(n+1) = A(n) x(n) $. Define $ H(n):= A^T (n) A(n)$. (a) Prove the Lagrange identity $$\displaystyle{ || x(n+1) ||_2 ^2 = x^T ...
5
votes
2answers
434 views

Can a symmetric matrix always be represented as the sum of a positive-definite and negative-definite matrix?

I was wondering if it is possible to decompose any symmetric matrix into a positive definite and a negative definite component. I can't seem to think of a counterexample if the statement is false. ...
6
votes
2answers
189 views

Prove that this matrix equation has no roots if a matrix meets certain conditions

Could you explain to me how to solve matrix equations? Here is an example: Prove that: $$2X^2 + X = \begin{bmatrix} -1&5&3\\-2&1&2\\0&-4&-3\end{bmatrix}$$ has no solutions ...
1
vote
1answer
185 views

Geometrical meanings of diagonalizable and normal matrices

We know that normal matrices are diagonalizable, but the converse is not true. For example, see here. Since a diagonalizable matrix represents a scaling operation under certain basis, so I wonder what ...
1
vote
1answer
44 views

Confusion in matrix manipulation

I have a vector $y = \frac{-x^TB}{C}$ Substituting y in $x^TAx + 2x^TBy+y^TCy$ I am supposed to get $x^T(A-BC^{-1}B^T)x$ I am just beginner with matrix stuff. Obviously, substituting y in the ...
1
vote
1answer
206 views

Rotating a line segment in 3D to a prescribed orientation

I have a general line segment with endpoints $(x_1,y_1,z_1)$ and $(x_2,y_2,z_2)$ referenced to a 3D Cartesian coordinate frame E. I wish to rotate this coordinate fram E to a new coordinate system F ...
1
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2answers
86 views

Show $e^{-a\sigma_3}\sigma_1e^{a\sigma_3} = \sigma_1e^{2a\sigma_3}$

How do you show that $$e^{-a\sigma_3}\sigma_1e^{a\sigma_3} = \sigma_1e^{2a\sigma_3}$$ where $\sigma_i$ are the Pauli matrices.
0
votes
1answer
84 views

Recovering a conic from a pole-polar pair

Consider a conic section $C$ in $\mathbb{R}^2$. Every point $P$ in the plane has a "dual" (pole-polar duality) line $L$ with respect to $C$ such that lines $PA$ and $PB$ are tangent to $C$, where $L ...
2
votes
0answers
44 views

Determinant of Jacobi of $C^3$ function

If I have a function $f:\mathbb{R}^n \to \mathbb{R}^n$ that is $C^3$, then the elements of the Jacobian matrix is $C^2.$ Is the determinant of the Jacobian matrix also $C^2$?
3
votes
1answer
97 views

Echelon form of matrix have the same rank, after operations modulo

In my homework task, I need to prove that if my matrix have rank $k$ modulo $\ell$, then it also have rank $k$ modulo $p$. Please give me advice, how to prove it. --edit-- The precise task is: On ...
1
vote
1answer
55 views

Linear transform of parameterization, but implicit?

Suppose I have a $2\times 2$ matrix $M$ in implicit equation form, where $(x,y)$ is transformed to $( xOut,yOut )$: $$ ax + by = xOut, \quad cx + dy = yOut $$ Now additionally I have a line $L$, also ...
4
votes
2answers
128 views

Where have I made my mistake in calculating $P^{-1}AP$?

I have $$Q = \begin{pmatrix} -\mu & \mu \\ \lambda & -\lambda \end{pmatrix}$$ and I want to work out the value of $\mathbb{P}(t) = \exp(Qt)$ So I diagonalised $Q$ and then worked out the ...
1
vote
1answer
106 views

For a first order inhomogenous system of linear differential equations, what is a good way of defining resonance?

I apologize for the title being slightly unclear (at least to me it seems so), so if anyone has a better suggestion feel free to change it. Anyways, for example, when dealing with a second order ...
1
vote
1answer
99 views

How to go from the diagonal matrix back to the matrix with the original basis?

I start off with matrix $Q$, such that $$Q = \begin{pmatrix} -\mu & \mu \\ \lambda & -\lambda \end{pmatrix}$$ and I want to work out the value of $P(t) = \exp(Qt)$ So to do this, you first ...
0
votes
1answer
141 views

Inequality for singular value for differences of matrices (upper bound)

Does anybody know the inequality of singular value for differences of matrices, i.e. $\sigma_{max}\left(\begin{array}{c} A-B\end{array}\right)\leq??$ in term of $\sigma_{max}\left(\begin{array}{c} ...
2
votes
3answers
254 views

Quadratic equation with matricial coefficients

If I have a equation in the form $${\lambda ^2}{I_N} + \lambda {M_1} + {M_2} = {0_N}$$ where ${I_N}$ is the identity matrix of order $N$, $M_1$ and $M_2$ are matrices of ($N\times N$) order and ...
3
votes
1answer
441 views

Find basis of im, ker and dim im, dim ker verification

In my homework, I've to find basis of im and ker of linear transformation $\varphi:\mathbb{R}^{3}\rightarrow\mathbb{R}^{2}, \varphi((x_{1},x_{2},x_{3}))=(2x_{1}+x_{2}-3x_{3},x_{1}+4x_{2}+2x_{3})$ My ...
5
votes
2answers
141 views

Calculate $\left\Vert \begin{bmatrix}1 & 2\\ 2 & 4 \end{bmatrix} \right\Vert$

With $$\left\Vert A \right\Vert=\max_{\mathbf{x}\ne 0}\frac{\left\Vert A\mathbf{x}\right\Vert }{\left\Vert \mathbf{x}\right\Vert }$$ and $$A=\begin{bmatrix}1 & 2\\ 2 & 4 \end{bmatrix} $$ ...
2
votes
5answers
207 views

Calculation matrix exponential

I got $$ A = \begin{pmatrix} 0 & \omega \\ - \omega & 0 \end{pmatrix}$$ with eigenvalues $\pm i\omega$ and eigenvectors $(-i,1)$ and $(i,1)$. Can I then calculate $e^{tA}$ by $$ e^{tA} = V ...
0
votes
1answer
103 views

How to Simplify this to the given answer? (matrix equation, trig functions)

Can someone help me to simplify $$\Bigg( \begin{matrix} 5\cos t &5\sin t \\2\cos t+\sin t & 2\sin t-\cos t\end{matrix} \Bigg) \Bigg( \begin{matrix} u_1 \\u_2\end{matrix} \Bigg) ...
3
votes
1answer
77 views

Dim E, set of linear transformations

Suppose $U\subset W \subset V$ are three linear spaces with respective dimensions 3, 6 and 10. Let $E\subset L(V,V)$ be the set of linear transformations $f:V\rightarrow V$ such that $f(U)\subset U$ ...
0
votes
1answer
223 views

What is a basis for the space of anti-symmetric $3\times 3$ matrices?

I tried to find a basis for the subspace of 3-by-3 anti-symmetric matrices - but for nothing. How to find such a basis?
7
votes
2answers
351 views

Maximal dimension of subspace of matrices whose products have zero trace

In the space of all real matrices with dimension $n$, find the maximal possible dimension of a subspace $V$ such that $\forall X,Y\in V,\, \operatorname{tr}(XY)=0$.
3
votes
3answers
512 views

Rank of matrix AB when A and B have full rank

Define $A$ as $m\times n$ matrix with rank $n$, and $B$ as $n\times p$ matrix with rank $p$. Calculate the rank of matrix $C=AB$. --edit-- Rank of a matrix is the number of linear independent rows.
3
votes
1answer
130 views

If $AB=0$, then $A+A^T$ or $B+B^T$ is singular

Define $A$ and $B$ as being square matrices of dimension $2011$. Prove that if $AB=0$, then at least one of matrices $A+A^{T}$ or $B+B^{T}$ have rank below $2011$. -- edit -- Rank of a matrix is ...
30
votes
3answers
1k views

Why, historically, do we multiply matrices as we do?

Multiplication of matrices — taking the dot product of the $i$th row of the first matrix and the $j$th column of the second to yield the $ij$th entry of the product — is not a very ...
7
votes
1answer
278 views

Prove that if $A^2x=x$ then $Ax=x$

I feel this should be easy but I cant solve this problem: Prove that if $A$ is a $n\times n$ matrix and $x$ a vector in $\mathbb R^n$ both with real positive entries and $A^2x=x$ then $Ax=x$. I ...
4
votes
1answer
85 views

Angle after two rotations in $\mathbb R ^3$

Question: A rotation through $45^{\circ}$ about the x-axis is followed by a similar one about the z-axis. Show that the rotation corresponding to their combined effect has its axis inclined at equal ...
3
votes
1answer
165 views

Diagonalizable vs Normal

If I have real valued matrix $A$, are these two notions of being Diagonalizable and being Normal equivalent?
1
vote
1answer
184 views

Eigenvector of A to given Eigenvalue which requires row swapping to get reduced echelon form

Given the Matrix $$A = \left(\begin{matrix} 0 & 1 & 0 \\ 0 & 0 & 1 \\ -1 & -3 & -3 \end{matrix}\right)$$ calculate the eigenvalues and the corresponding eigenvectors of ...
-2
votes
2answers
333 views

How do I find the angle $\theta$ of the line through the origin that matches the given points the best?

How to find angle $\theta$, that the line passing through the origin that is the best fit for the points given below in the mean square sense makes with the horizontal axis. $$x_1=[1\;\; 2]^T$$ ...
0
votes
0answers
97 views

Can a matrix based “secret sharing scheme” be applied to image based secret sharing?

I was reading this and was wondering if it can be used to do secret sharing with images. I dont know a lot about image processing, but if the authors have given a scheme for matrices, how can be ...
5
votes
0answers
934 views

Possible Jordan Canonical Forms Given Minimal Polynomial

I was supposed to find all possible Jordan canonical forms of a $5\times 5$ complex matrix with minimal polynomial $(x-2)^2(x-1)$ on a qualifying exam last semester. I took the polynomial to mean ...
0
votes
1answer
112 views

Minimization to Maximization doubt in SVM

I came across a lecture on Support Vector Machines and in the lecture they converted a maximization problem into a minimization problem. I am wondering how it was done... $ Max \frac {1}{||x||} $ ...
2
votes
2answers
139 views

prove that a linear map is injective - $T(X) = X + 2X^T$

I have the following linear map: $$T: \operatorname{Mat}_{n\times n}(\mathbb{R}) \to \operatorname{Mat}_{n\times n}(\mathbb{R})\;,$$ $$T(X) = X+2X^T\;.$$ I have to prove that it is injective ...
6
votes
3answers
179 views

Simplicity of eigenvalue

I have a matrix $A$ and I introduce $(I+A)^{m},$ where $I$ is the identity matrix of same order with $A$ and $m$ is a positive integer. I want to show that if $(1+ \lambda )^m$ is a simple eigenvalue ...
-1
votes
3answers
274 views

Why a full-rank matrix in a finite field is also full-rank in a expanded finite field?

For example, matrix MAT is full-rank in $GF(2^8)$, why MAT is also full-rank in $GF(2^{16})$ and $GF(2^{32})?$ Thanks
2
votes
1answer
62 views

$GL(2,\mathbb{R})$ as a subset of $\mathbb{R}^4$

If we consider $GL(2,\mathbb{R})$ as a topological subspace of $\mathbb{R}^4$ with the usual topology and want to know if it compact or not then if we could show that it was not closed then we would ...
2
votes
1answer
134 views

Relation between singular values of matrices and their products

Is there any explicit relation between the singular values $\lambda_X$ and $\lambda_Y$ of two same size matrices $X$ and $Y$, respectively, and the singular values $\lambda_{XY^t}$ of the matrix ...
0
votes
1answer
42 views

how to prove $\|(A^HA)^k\| =||A||^{2k}$ using singular value decomposition

how to prove $\|(A^HA)^k\| =||A||^{2k}$ using singular value decomosition. $A^H$ is a hermitian matrix. $A$ element of $C^{p\times q}$, for every positive integer $k$.