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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Determinant of 5x5 matrices

Let A and B be 5x5 matrices with det(-3A)=4 and det(B^-1)=2. Find the det(A), det(B) and det(AB). My answer : det(A)=-12 , det(B)=1/2 and det(AB)=-6. Wish to check my answer, thank you.
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Negative definiteness of a block matrix

There is a block matrix, \begin{align} M=\left(\begin{array}{cc} A & B\\ C & I \end{array}\right)\quad\text{where}\quad A<0 \end{align} I am curious whether the matrix $M$ is negative ...
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Is it possible to get the original eigenvector after scaling a matrix?

Let ${\mathbf{X}}\in\mathbb{R}^{n\times 1}$ and ${\mathbf{Y}}\in\mathbb{R}^{n\times 1}$ and let $\mathbf{A}\in\mathbb{R}^{n\times 2}$ be defined as \mathbf{A} = \left[{\mathbf{X}}|{\...
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Constructing an eigenvector for a certain matrix representing a graph with a perfect code

Let $A$ be a symmetric $(0,1)$-matrix whose row sum is $r.$ Suppose I have a $(0,1)$ vector $v$ such that $$Av = \vec{1} - v.$$ By taking the vector $$u = \vec{1} - (r+1)v$$ we see that Au = A (\...
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Recurrence Relations for Sequence Counting Hamming Weights

Define $a(n,k)=|\{x \in \mathbb{F}^{2n}_2 : Hx=0, ||x||=k\}|$ and $b(n,k)=|\{x \in \mathbb{F}^{2n}_2 : Hx=1, ||x||=k\}|$ where $||\cdot||$ denotes the Hamming weight of $x$ (i.e. number of non-zero ...
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How to divide by a matrix

I found a question in an old exam, where the function $\phi(z) := \frac{\exp(z) - 1}{z}$ is given. Now we evaluate $\phi(\mathbf{A})$. But how do I divide by a matrix? I already thought about ...
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What does $M_{uv}^l$ represent?

Let $M$ be any non negative square matrix. What does $M_{uv}^l$ represent? $M_{uv}^l$: $uv$ entry of $M^l$. (When $A$ is adjacency matrix of a graph, then $A_{uv}^l$ is number of walks of length $l$ ...
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Equivalent definitions of an orthogonal matrix.

I wish to show that the following definitions of an $n \times n$ real matrix $Q$ are equivalent: $QQ^T=I$, $Qx\cdot Qx=x\cdot x$ for all $x\in \mathbb{R}^n$. I found it easy to show that ...
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Technical question in Vandermonde determinat proof

I can follow the proof given in (2nd proof, or the induction proof), until the sentence: "From the Expansion Theorem for Determinants‎, we can see that the coefficient of $x_k$ is:". I don't ...
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Showing that the set of $2 \times 2$ real orthogonal matrices has a particular parameterization [duplicate]

Theorem Every orthogonal matrix in $\mathbb{R}^{2, 2}$ is in the form \begin{bmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{bmatrix} or \begin{bmatrix} \cos\theta &...
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Is it possible to determine if a matrix is not diagonalizable via row operations?

Suppose a matrix can be row reduced to the identity matrix, is this enough to say that it is not diagonalizable? If so, what theorem(s) or logic figures this out?
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Simple question - represent vector with respect to a basis

Basic question here, I've always been weak at this stuff. Suppose that we have a situation $U=WX$ where $U,W,X$ are matrices that are known to us. You can suppose that $U$ is invertible. I want to ...
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Matrix of perspective?

I worked with matrices of translation, rotation, scale but the perspective one is new for me. Could you explain why there are a lot of examples of perspective matrices with various values inside? For ...
Let $A,B \in M_n$, $n \geq 2$. If $A$ and $B$ do not share a common eigenvector, why is $\mathcal{A}(A,B) = M_n$? Notation and definitions: $M_n$: the set of $n \times n$ matrices over \$\mathbf{C}^...