# Tagged Questions

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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### Inverting Matrix Equations Y = F(X)

I have a arbitrary $N \times N$ matrix $S$. I have an function of this matrix given by: $$A = F(S) = 2S + P^{-1}S + 2SP + PSP$$ where $P$ is the a cyclic permutation matrix which when acting on the ...
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### Similarity between special matrices and special complex numbers

From Wikipedia: It is occasionally useful (but sometimes misleading) to think of the relationships of different kinds of normal matrices as analogous to the relationships between different ...
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### Fast Algorithm For Adding An Equation To A System?

Assume an $N \times N$ matrix $A$ and a length $N$ vector $b$. I've already solved the system $Ax = b$ for $x$ using standard methods. (If you want you can assume that I have the inverse of $A$ as ...
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### Wikipedia Article — Legendre Transform

I was reading the wiki article on Legendre Transform. I would be grateful if someone could explain the section at http://en.wikipedia.org/wiki/Legendre_transformation#Examples ie how they arrived at ...
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### Looking for “average” of two permutations

I am a computer programmer and I am building a search engine for a client. Right now I am puzzling myself about the order in which I should return search results. There are two obvious orderings: ...
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### Linear Algebra: An explanation on a simplification

Could someone please explain to me what property was used in simplifying this, or how this was achieved? Thank you.
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### Solving a matrix equation $AX=XB$ in a CAS

I have the following computational problem. Let $N$ be a positive integer and $A\in \mathbb{C}^{2N\times 2N}$, $X\in \mathbb{C}^{2N\times 4}$ and $B\in \mathbb{C}^{4\times 4}$. I want to solve the ...
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### How to 'minimize' correlation between series

Hi fellow mathemagicians, let's say that I have 3 series of numerical results (they represent 'drawdowns') : ...
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### Is this a positive semi- definite matrix

I have a matrix $A$, which satisfies : $A$ is symmetric; all the diagonal entries of $A$ are equal to $1$; other entries of $A$ is between $0$ and $1$. My question is, whether $A$ is a positive ...
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### For every integer $n>1$ , does there exist a diagonal matrix $D \in M(n,\mathbb R)$ such that $AD=DA$ holds only if $A$ is diagonal?

Is it true that for every integer $n>1$ , there exist a diagonal matrix $D \in M(n,\mathbb R)$ such that $A \in M(n,\mathbb R)$ and $AD=DA \implies A$ is also a diagonal matrix ?
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### Gaussian elimination algorithm performance

I am developing the quadratic sieve algorithm and I reached a new bottle neck: The matrix processing. I been reading quit a lot about this topic and I found many solutions Gaussian elimination: ...
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### Finding Eigenvectors for $3 \times 3$ matrix with rows of zeros.

For a $3 \times 3$ matrix: [A]$= \begin{bmatrix} 6 & -2 & 2 \\ -2 & 3 & -1 \\ 2 & -1 & 3 \end{bmatrix}$ I have the eigenvalues: ...
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### 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, ...
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### Is every “weakly square” matrix either a $0$ matrix, or a square matrix?

Call a matrix $A$ weakly square iff $\mathrm{det}(A^\top A) = \mathrm{det}(A A^\top)$. Then clearly, every square matrix is weakly square, and every zero matrix is weakly square. Question. Are ...
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Let $U_n$ be the upper triangular Pascal matrix, $L_n$ the lower triangular Pascal matrix of n-th degree, i.e. $$u_{ij} = \begin{cases} \binom {j-1}{j-i} & \quad i\le j\\ 0 & ... 2answers 61 views ### If a matrix has a unique left inverse then does it necessarily have a unique right inverse (which is the same inverse)? I know if a matrix has a left and right inverse then the inverses are the same and are (is) unique and the original matrix is a square matrix, thus if I have a matrix which has multiple left inverses ... 1answer 27 views ### A>0 , \sum\limits_{i = 1}^n {{x_i}} = 1 , Ax = \rho (A)x ,can we say that \rho (A) = \sum\limits_{i,j = 1}^n {{a_{ij}}{x_j}} ? Let A>0(i.e, all a_{ij}>0). There is x>0(i.e, all x_{i}>0) such that: \sum\limits_{i = 1}^n {{x_i}} = 1 Ax = \rho (A)x Can we say that \rho (A) = \sum\limits_{i,j = ... 1answer 21 views ### If Ax=c(x)e, \forall x, then A has rank one How to prove if Ax=c(x)e, \forall x, then A has rank one? e is a vector with all entries one. c(x)\in \mathbb{R}, which is a constant depending on x My method is by Gaussian ... 1answer 41 views ### The trace of the product of positive definite matrix with the inverse of itself plus another positive definite matrix Let A be an n\times n positive definite matrix with eigenvalues a_1,a_2,\dots,a_n in descending order. Let T be an n\times n positive definite matrix with eigenvalues in t_1,\dots,t_n in ... 2answers 28 views ### Simultaneous diagonalization Given two symmetric matrices A,B\in\Bbb R^n how can we find if they are simultaneously diagonalizable? If they have such property how can we find U such that UAU' and UBU' are simultaneously ... 1answer 26 views ### Diagonalizable transmit to submatrix If$$\begin{pmatrix} A & B\\ \Large 0 & C \end{pmatrix}$$is similar to a diagonal matrix, are A and C also similar to diagonal matrices? 1answer 28 views ### Finding the formula for a linear transformation given the transformation of the basis vectors. Consider the basis \{\vec{p},\vec{q}\} where \vec{p}=(1,1) and \vec{q}=(-1,0). Let T:\mathbb{R}^2\to\mathbb{R}^2 be the linear operator such that T(\vec{p})=(1,-2) and ... 1answer 45 views ### Questions about invertible matrices… If I have two n \times n matrices M,N that have an inverse, how do I show that: {M^t} has an inverse with {({M^t})^{ - 1}} = {({M^{ - 1}})^t} that if M and N are orthogonal ({M^{ - 1}} ... 1answer 54 views ### Finding the inverse of a “bow-shaped” matrix Consider the matrix$$A = \begin{bmatrix} n_{+} & n_1 & n_2 & n_3 & \cdots & n_{r-1} \\ n_1 & n_1 & 0 & 0 & \cdots & 0 \\ n_2 & 0 & n_2 ...
The question asks WHY is it true that $$A^{n} = PD^{n}P^{-1}$$ I can never do proper proving in algebra; what I almost know for sure is that a proof by induction is the way to go here. But how do you ...
### Matrix ODE, defective eigenvalue: Where does the extra '$t$' come from?
Given $A \in \Bbb R^{2\times 2}$, the system $$\dot X=AX$$ Has the solution $$X= c_1e^{\lambda t}\xi_1+c_2e^{\lambda t}\left (\xi_1 t+\xi_2 \right)$$ Where $\xi_1$ is the unique ...