# Questions tagged [matrices]

For any topic related to matrices. This includes: systems of linear equations, eigenvalues and eigenvectors (diagonalization, triangularization), determinant, trace, characteristic polynomial, adjugate and adjoint, transpose, Jordan normal form, matrix algorithms (e.g. LU, Gauss elimination, SVD, QR), invariant factors, quadratic forms, etc. For questions specifically concerning matrix equations, use the (matrix-equations) tag.

38,079 questions
Filter by
Sorted by
Tagged with
57 views

### Prove $Ax = \frac{1}{2}x$ only has the trivial solution where $A$ has all integer entries.

Prove $Ax = \frac{1}{2}x$ only has the trivial solution where $A$ is a $n \times n$ matrix with integer entries and $x = (x_1, \ldots , x_n)$. I am a bit rusty on my linear algebra and trying to ...
25 views

### Convert from one matrix to another

I have this 4x4 matrix A: a, b, c, 0, e, f, g, 0, i, j, k, 0, 0, 0, 0, 1 And I want to convert it to this matrix B: ...
38 views

14 views

39 views

### Finding limit using Markov chain.

Here is the question: I set up the probability matrix as: \begin{bmatrix} 0 & 1/3 & 1/3 & 1/3 \\ 0 & 1/2 & 1/2 & 0 \\ 1 & 0 & 0 & 0 \\ 1/2 & 0 & 1/2 &...
81 views

### Proof of the matrix identity $\det\begin{pmatrix}A&B\\B&A\end{pmatrix}=\det(A+B)\det(A-B)$

The Wikipedia page about the determinant mentions the following matrix identity $$\det\begin{pmatrix}A&B\\B&A\end{pmatrix}=\det(A+B)\det(A-B),$$ valid for squared matrices $A$ and $B$ of the ...
81 views

### Rank-1 update eigenvalues

If I had a diagonal matrix with rank-1 update $$D + uv^T$$ what can I say about its eigenvalues? I know from Two matrices that are not similar have (almost) same eigenvalues that every eigenvalue ...
37 views

### $\frac{1+m_v}{1+m_u}\leq \frac{1+u^T(M+I)^{-1} u}{1+v^T(M+I)^{-1}v} \leq \frac{1+m_u}{1+m_v}$ if $M$ is positive sym. PD & $u,v$ are $0-1$ vectors?

Let $n$ be a positive integer. Let $m_u,m_v \in \{1,...,n-1 \}$. Let $M$ be a $n \times n$ symmetric positive definite matrix with positive entries. Let $u$ and $v$ be vectors of length $n$ with ...
How can you prove that $\mathrm{E}\left\lbrace \mathbf{x}(k) \mathbf{x}^{T}(k) \mathrm{E}\left\lbrace \Delta \mathbf{w} (k) \Delta \mathbf{w}^{T} (k)\right\rbrace \mathbf{x}(k) \mathbf{x}^{T}(k) \... 2answers 48 views ### How to solve$M - EV - (EV)^T = 0$against$E$? I have a matrix equation: $$M - EV - (EV)^T = 0$$ which i want to solve against$E$. Is this possible? How to do that? Remarks: matrices are square,$E$,$M$are symmetric,$V$is invertibile. ... 3answers 56 views ### How to find the set of solutions of$Ax=By$for$A,B$matrices? Let$A,B$be arbitrary matrices with the same number of rows. How can we find the set of solutions$x,y$to the matrix equation$Ax=By$? I understand that this problem is probably related to that of ... 1answer 22 views ### Matrix calculations in second order polynomial approximation I am looking at second order polynomial approximation, specifically at this link. However, I am stuck in the one dimensional case: I do not think I can calculate the following: $$X^{T}*X^{-1}*X^{T}... 0answers 62 views ### Why are operators often written in 3s? In a lot of my quantum mechanics and linear algebra books, operators are often defined as M=IMI and many operations on operators are often done in similar fashion, for eg. operator M might be ... 1answer 89 views ### Proof rank linear map is equal to the rank of its transformation matrix I want to prove, that the rank of a linear map f: V \rightarrow W is equal to the rank of the transformation matrix A of this linear map. Let v a Basis of V with length n and w a Basis of ... 1answer 56 views ### Is there an easy way to simplify (P+Q)^2 - (P-Q)^2 where P and Q are matrices? P and Q are different matrices that are 2x2. I have tried to do this by letting both equal a 'made-up' matrix with pro numerals, but the expansion process is very long. Is there an easy way to ... 2answers 71 views ### When are the eigenvalues of a positive-definite matrix \le 1? The eigenvalues of a positive-definite matrix are guaranteed to be > 0; but does anyone know of sufficient conditions when they will also all be \le 1? 1answer 49 views ### If A\in {\mathbf F_2}^{n\times n} is symmetric, its diagonal is in the span of its columns Let A be an n\times n symmetric matrix with entries in \mathbf F_2 (the field with 2 elements, also referred to as \mathbb Z/2\mathbb Z). Prove that the diagonal of A is in the span of its ... 1answer 80 views ### Diagonalization of a symmetric matrix I'm trying to prove that every symmetric matrix is diagonalizable. I know that there are already many answers regarding this question, but I just want to check out whether my approach is correct or ... 0answers 11 views ### Matrix perturbation and Eigenvector Bound Recently I've been learning Matrix perturbation theory. Many theorems deal with perturbation bound in l_2 norm, or more generally, unitarily invariant norm, like F-norm or others(Like Davis-Kahan's ... 3answers 37 views ### similar matrices and one-to-oneness of matrix transformation I need help with this proof Suppose A and B are n \times n matrices and that there is an invertible matrix P such that A=PBP^{-1}. Show that if x \mapsto Bx is one-to-one, then x \mapsto ... 1answer 46 views ### Showing u^T (M+I_n)^{-1} u \geq v^T(M+I_n)^{-1}v when M is symmetric PD with positive entries and u,v are 0-1 vectors Let n be a positive integer. Let m_u,m_v \in \{1,...,n-1 \}. Let M be a n \times n symmetric positive definite matrix with positive entries. Let u and v be vectors of length n with ... 3answers 56 views ### Showing u^T M u \geq v^TMv when M is symmetric PD and u,v are 0-1 vectors Let M be a n \times n symmetric positive definite matrix. Let u and v be vectors of length n with entries consisting n-m_u (or n-m_v) 0's and m_u (or m_v) 1's, where m_u,m_v \... 0answers 12 views ### Why does Matrix Factorization (like ALS) lead to good results for missing values? As the title already suggests, I'm not really understanding WHY Matrix Factorization methods like Alternating Least Squares (ALS) lead to good predictions of missing values. As far as I understand we ... 1answer 26 views ### Linear Algebra - Find an orthonormal basis Question: Consider the space of 2 \times 2 matrices equipped with the inner product: \langle A , B \rangle = \operatorname{tr}(A^T B) i) Find an orthonormal basis for the subspace V = \{M \... 1answer 46 views ### Proving that symmetric matrix is diagonalizable I'm trying to prove that symmetric matrix (with real entries) is diagonalizable. Here, Ittay Weiss proved the result. I'm following with his argument, but I couldn't properly understand the following ... 1answer 47 views ### Prove that the spectrum of normal matrices is stable under small perturbations Let A be a normal matrix. As shown for example in these notes (link to pdf, see pagg. 120 and 121), given a matrix B with a single non-zero entry and some parameter \varepsilon\in\mathbb R, the ... 1answer 66 views ### Finding the Gradient Matrix for the given expression Let \rho be a matrix, and let \rho_A be the partial trace of the matrix \rho. For simplicity, let us assume \rho is a 4 \times 4 matrix. The partial trace is defined as follows: If$$\rho ... 0answers 23 views ### Calculating determinant of integral of matrices I would like to calculate the determinant of some$3\times3$matrix $$\boldsymbol{A}(t) = \int_{0}^{t}\boldsymbol{B}(s-t)\boldsymbol{C}(s-t)\boldsymbol{D}(s-t)ds$$ in terms of the determinants of the$...
Let $A$ be a general $n\times n$ invertible matrix. Let $T^A$ be the "transposer" matrix i.e. $T^A A = A'$. (Does that $T^A$ multiplied by $A$ equal the transpose of $A$?) Then does $T^A$ depend on ...