# 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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### Compact representation of the following matrix

I have a matrix that has the following structure \begin{bmatrix} a_1(1) & 0 & a_2(1) & 0 \\ 0 & a_2(1) & 0 & a_1(1) \\ a_1(2) & 0 & a_2(2) & 0 \\ 0 & a_2(2) &...
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### Show that the diagonal entries of symmetric & idempotent matrix must be in [$0,1$]

Show that the diagonal entries of symmetric & idempotent matrix must be in [$0,1$]. Let $A$ be a symmetric and idempotent $n \times n$ matrix. By the definition of eigenvectors and since $A$ is ...
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### Desk Reference Question: Linear Algebra and its Applications by Lay or Strang? Or Handbook of Linear Algebra?

I would like a good working desk reference for linear algebra for someone in applied mathematics (no proving abstract theorems). I saw the Handbook of Linear Algebra as well, but I am concerned it may ...
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### How to determine if matrices are similar?

Trying to teach myself some Linear Algebra, now trying to study about similar matrices concept, but i am having some trouble (maybe because i am trying to teach myself), found a question online and i ...
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### Prove that $|GL_n(\mathbb{F})|< q^{n^2}$.

Let $\Bbb F$ be a finite field, say $|\Bbb F|=q$; then we know that $|GL_n(\Bbb F)| < \infty$. But how can we prove that $|GL_n(\mathbb{F})|< q^{n^2}$? I'm guessing because there $n^2$ ...
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### Algorithm for real matrix given the complex eigenvalues

Given complex eigenvalues (occurring in conjugate pairs) how to get a single instance of a real matrix which has these eigenvalues. I know the matrix is not unique as eigenvectors are not fixed but in ...
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### updating of the cholesky decomposition

I try cholrank1 update (wikipedia) of the symmetric positive definite (SPD) matrix . ...
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### Constructing matrices with eigenvalues equal to roots of a given polynomial

Suppose we are given a polynomial e.g. $$x^4+Ax^3+Bx^2+Cx+D,\tag1$$ and we need to construct a matrix, whose eigenvalues would be equal to roots of this polynomial. One way, rather inelegant, is to ...
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### prove that there exists an upper triangular matrix U such that (U^T)U=A

Let A be a positive definite matrix \begin{pmatrix} a & b \\ b & c \\ \end{pmatrix} prove that there exists an upper triangular matrix U such that U transpose times U equals A. I'm ...
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### Is this operator $A = \pmatrix{1&1\\0&1}$ self-adjoint?

Is this operator $$A = \pmatrix{1&1\\0&1}$$ self-adjoint? I think not, because $$\pmatrix{1&1\\0&1}^T\neq A$$ where $T$ is the transposition of the matrix. What do you all think?
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### Rotate a unit sphere such as to align it two orthogonal unit vectors

I have two orthogonal vectors $a$, $b$, which lie on a unit sphere (i.e. unit vectors). I want to apply one or more rotations to the sphere such that $a$ is transformed to $c$, and $b$ is transformed ...
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### $A$ is a doubly stochastic matrix, how about $A^TA$

I am reading a paper with assumption that $A \in R^{n\times n}$ is a doubly stochastic matrix. However, the paper says $A^TA$ is symmetric and stochastic. Since $A^TA$ is symmetric, if $A^TA$ is ...
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### Eigenvalues of Householder matrix

What would be the eigenvalues for a Householder matrix defined as: $H = I - 2 u u^T$? Can someone explain it to me intuitively or with a simple proof?
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### Determine the dimension of $U+W$ and of $U \cap W$. Which sums are direct sums?

Problem: Determine the dimension of the sum $U + W$ and of the intersection $U \cap W$ of the following subspaces $U$ and $W$. Which sums are direct sums? 1) $U = \text{span}\left\{(1,1,1)\right\}$ ...
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### Characterization of positive definite matrix with principal minors

A symmetric matrix $A$ is positive definite if $x^TAx>0$ for all $x\not=0$. However, such matrices can also be characterized by the positivity of the principal minors. A statement and proof can, ...
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### Why the Householder matrix is orthogonal?

A Householder matrix $H = I - c u u^T$, where $c$ is a constant and $u$ is a unit vector, always comes out orthogonal and full rank. Why $H$ is orthogonal (looking for an intuitive proof rather than ...
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### Getting VAR parameters from a research paper.

Many econometrics papers provide the parameters used in their VAR model. If I notate my VAR model as $$z_{t+1} = c + B z_{t} + \Sigma \epsilon_{t+1}$$ where $\epsilon \sim N(0, I)$, then I need to ...
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### Hesse matrix is negatively semidefinite if a function has a local maximum

Let $U \subset \mathbb{R}^n$ be open, and let $f: U \to \mathbb{R}$ be at least twice continuously differentiable. Also, we assume $f$ has a local maximum $a \in U$. I now want to show that the Hesse ...
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### Is there a way to turn this summation into a matrix multiplication?

I have two vectors $\mathbf{s}, \mathbf{p}$ of length $n$, and I need to compute a vector $\mathbf{\pi}$ defined by $$\pi_i=\sum_{j=1}^is_j(p_j-p_i)$$ for $i$ from $1$ to $n$. I suspect this ...
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### For certain positive semidefinite matrices, subtracting the outer product of their row-sums does not change the positive semidefiniteness

Let $e$ denote the vector of all ones, $J=ee^T$ and $\langle A,B\rangle = trace(AB^T)$. Consider a symmetric positive semidefinite (psd) matrix $A\geq 0$ (that is, $a_{ij} \geq 0$ for all entries) ...
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### Number of components needed for 3D rotation

Using Euler angles, a 3D rotation can be expressed using 3 real numbers. Using quaternions, 4 are needed and using rotation matrices 9. Is it possible to express a 3D rotation using less than 3 real ...
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### How to prove infinite solution vs no solution for singular matrix problem.

In the problem Ax=B My coefficient matrix is \begin{bmatrix} α-1 & 1-α\\ α & -α \end{bmatrix} x is \begin{bmatrix} ln K\\ ln L \end{bmatrix} b ...
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### generating orthogonal parity check matrix, from a random generator matrix

I have a matrix G of dimension 13x20. It is a full rank matrix. It is not in the standard form of a generator matrix. Now for the parity check matrix 'H', I need a standard representation H=[-P;I]. ...
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### Matrix with entries equal to $1$ and $-1$ (Sign Matrix)

What can we say about the determinant and (or) maximum eigenvalue of a matrix with entries equal to $1$ and $-1$. Further assume that the rows and columns are linearly independent. Are there special ...
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### How do I solve for vector $P$ in the matrix equation $s=A'B^{-1}A$?

I would like to rearrange the matrix equation $s=A'B^{-1}A$ into the form $A=f(s,B)$ (i.e., some function of $s$ and $B$), where s is scalar, $A$ is $n\times 1$, $A'$ is the transpose of $A$, and $B$ ...
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### Is there an explicit formula for $\left(xx^T\right)^{-1}$ with $x\in\mathbb{R}^n\setminus\left\{0\right\}$?

Let $x\in\mathbb{R}^n\setminus\left\{0\right\}$. Obviously, $$A:=xx^T$$ is symmetric and positive definite. Hence, $A$ is invertible. Can we find an explicit formula for $A^{-1}$?
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### Proof that for all symmetric matrices $A$ and $B$, $AB=(BA)^T$.

Recall that a matrix, $M$, is said to be symmetric if and only if $M=M^T$. I've been trying to use the homomorphic nature of the transpose operator to prove this proposition but this approach hasn't ...
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### Differentiating quadratic form containing vector raised to powers elementwise, can I avoid Hadamard notation?

Say $\mathbf{M}$ is a symmetric, p.d. 2x2 matrix, and $\mathbf{x}$ is a 2x1 vector. The familiar quadratic form is of course given by: $A=\mathbf{x'}\mathbf{M}\mathbf{x}$ (where $A$ is a scalar), and ...
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### How to determine a basis and the dimension for this vectorspace?

Determine a basis and the dimension for the following vectorspace: \begin{align*} W = \left\{A \in \mathbb{R}^{3 \times 3} \mid A \ \text{is a diagonal matrix and} \ \sum_{i=1}^3 A_{ii} = 0\right\} \...
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### Checking if a matrix is in the span of other matrices

Problem: Expand the following set matrices \begin{align*} \left\{ \begin{pmatrix} 1 & -1 \\ 2 & 3 \end{pmatrix}, \begin{pmatrix} 2 & 1 \\ -1 & 4 \end{pmatrix}, \begin{pmatrix} 0 & -...