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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12
votes
4answers
339 views

Converting recursive equations into matrices

How do we convert recursive equations into matrix forms? For instance, consider this recursive equation(Fibonacci Series): $$F_n = F_{n-1} + F_{n-2}$$ And it comes out to be that the following that ...
7
votes
5answers
117 views

Positive semi-definite of a matrix composed of semi-definite blocks

Say a matrix A is positive semi-definite. Let B be a square matrix composed of replicas of A as sub-blocks, s.t. $$B=\begin{pmatrix} A & A \\ A & A \\ \end{pmatrix},$$ or $$\begin{pmatrix} A ...
1
vote
1answer
37 views

Constructing regular integer matrices with distinct integer eigenvalues

How can I construct matrices with positive integer values and distinct integer eigenvalues (not necessarily positive, but 0 should not be an eigenvalue). The standard-method to construct matrices ...
1
vote
1answer
29 views

Triangularisation of a linear transformation

I understand that Upper triangular matrices must have at least one eigenvector, but why does this mean that the basis of $[T]_B$ must contain an eigenvector for $[T]_B$ to be upper triangular?
7
votes
2answers
101 views

Proof of the inequality $\sqrt{\det X} \leq \frac{\operatorname{tr}X}{2}$

Let $A, B \in M_2(\mathbb{R})$ be symmetric and positive definite. Put $X:=AB$. then, we have the following inequality: $$\sqrt{\det X}\leq \dfrac{1}{2}\operatorname{trace}X.$$ and the equality ...
1
vote
0answers
35 views

Matrix multiplication in quaternions is not necessarily linear

I tried to show by example that matrix multiplication for quaternionic matrices is is not necessarily $\mathbb H$-linear. If $A \in M_n(\mathbb H)$ is a quaternionic matrix and $x$ is a vector in ...
1
vote
0answers
36 views

Schur decomposition of real-eigenvalue matrix

Is Schur decomposition of real-eigenvalue matrix a real orthogonal decomposition? If yes, why is it? Is it because all the eigenvectors are real? If I have $$ A^T+A^2=I $$ then I deduced ...
1
vote
1answer
44 views

$A,B$ matrices , prove $Bv = \Lambda v$

$A,B$ are $n \times n$ matrices and $AB = BA$ Also, there is an eigenvalue $\Lambda$ in $A$ which its geometric complexity is $1$. Also there is $ v \ne 0 $ that $v$ is an eigenvector of $A$. ...
-1
votes
0answers
29 views

Order of eigenvectors in jblas?

I am using jblas to compute eigenvectors of a double symmetric matrix. Using symmetricEigenvectors(myMatrix)[0], I can get a matrix which columns are the eigenvectors of my matrix. However I need them ...
3
votes
2answers
120 views

What is the intutive explanation of why the notation of matrices is as it is?

If I want to solve a system of linear equations, like 2x-y=1 x+2y=4 Then the matrix notation for the same would be: $$ \begin{bmatrix} 2 & -1 \\ 1 & 2 \\ \end{bmatrix} \begin{bmatrix} X\\ ...
0
votes
1answer
37 views

Show that $\lambda_1 = \min \{ Q(u) \mid \|u\| = 1 \}$ and $\lambda_m = \max \{ Q(u) \mid \|u\| = 1 \}$

Let $V$ a vector space over $K$ and $Q(u) = \langle u, Tu \rangle$ a quadratic form. $T$ is a symmetric operator. The eigenvalues of $T$ are sorted by size $\lambda_1 < \dots < \lambda_m$. How ...
3
votes
1answer
45 views

If matrix $A$ is invertible, then there is a permutation of its rows leaving no-zeros on the diagonal

I need to prove this statement: "If $A$ invertible, then exist a permutation of its rows leaving no-zeros on the diagonal" and I tried using the definitos of invertible matrices and $LU$ ...
2
votes
1answer
52 views

When does $x^TAx + c^Tx$ have a global minimum?

This question is closely related to my last question about extended quadratic forms. I figured out a nice criterion, when $$f : \mathbb R^n \rightarrow \mathbb R$$ $$f(x) = x^TAx + c^Tx$$ has a ...
2
votes
3answers
50 views

Proving linear independence of matrices

Let $A = \textrm{diag}(a_{1},a_{2},a_{3})$ where $a_{1},a_{2},a_{3}$ are distinct. I am trying to show that every diagonal $3\times3$ matrix cane be made up of linear combinations of $I$, $A$ and ...
0
votes
1answer
23 views

Does cofactor expansion generalize to complex matrices?

When finding the determinant of some $n * n$ matrix $A$ when $$\forall i,j\in\mathbb{N} ,i\leq n\land j\leq n\implies A_{ij} \in \mathbb{C}$$ Can cofactor expansion be used under the normal definition ...
0
votes
1answer
15 views

Proof of product of symmetric matrices

Use the inverse of a 2x2 matrix formula to confirm that: "The product of two symmetric matrices is symmetric if and only if the matrices commute." IN THE CASE where the symmetric matrix ...
0
votes
1answer
16 views

covariance matrix in multivariate Gaussian distribution (semi-positive or positive definite)

The book that I'm reading states clearly that the covariance matrix $\Sigma$ in the following equation is a positive semidefinite and symmetric matrix. $$ p(x) = det(2\pi\Sigma)^{-1/2} exp\{ ...
2
votes
2answers
202 views

Newly Developed With Details - Describing orthographic projection using simple 2D transformations

Thanks to Pedro for helping me further develop my question into something tangible. His (most recent) answer below clearly and formally outlines what I am asking. This is similar to this question, ...
1
vote
2answers
108 views

$A^2$ is diagonalizable leads to $A$ diagonalizable?

If $A^2$ is diagonalizable, is it necessary true that $A$ is diagonalizable? Also, the opposite: If $A$ is diagonalizable, is it necessary true that $A^2$ is diagonalizable? I'm not sure yet, tried ...
-1
votes
1answer
29 views

Prove the sum of symmetric and non-symmetric matrix is $M^R_{2\times2}$

Prove that: $M^R_{2\times2} = \{A \in M^R_{2\times2} | A^t = A\} \oplus \{A \in M^R_{2\times2} | A^t = -A\}$ Well, it's pretty obvious that I need to show that the sum of symmetric and non-symmetric ...
12
votes
2answers
603 views

Properties of 4 by 4 Matrices

Define $ A=\begin{pmatrix} x_1 & x_2 & 0 & 0\\ 0 &1&0&0\\ 0&0&1&0\\ 0&0&0&1 \end{pmatrix}, B=\begin{pmatrix} 1 & 0 & 0 & 0\\ x_3 ...
1
vote
1answer
43 views

Meaning of the characteristic polynomial of a matroid

From wikipedia The characteristic polynomial of a matroid $M$ (which is sometimes called the chromatic polynomial,[29] although it does not count colorings), is defined to be $$ p_M(\lambda) ...
7
votes
5answers
357 views

Understanding matrices.

I'm trying to understand matrices. As far as I can understand, a matrix is a way to represent data (?) or some sort of function on data (?). However apart from the fact that they're a way to ...
1
vote
0answers
85 views

Eigenvalues of $\pmatrix{1&1\\1&2}$

I use maxima for calculation eigenvalues of this matrix: $$ \begin{bmatrix} 1 & 1 \\ 1 & 2 \end{bmatrix} $$ and I get $\frac{3\pm\sqrt{5}}{2}$ and then $[1,1]$ for some reason. Namely: ...
0
votes
0answers
28 views

SVD (Singular value decomposition) for matrix with constant row sum and column sum

M is a m*n matrix, the row sum equals to (K/m) and the column sum equals to (K/n). After SVD, we can find three matrix U,S,V, M=USV where U,V are orthogonal matrix and S is diagonal matrix. I can ...
0
votes
2answers
66 views

Is the determinant of a matrix some kind of “integral” of the linear mapping?

A $n \times n$ matrix corresponds to a linear mapping between two $n$-dim vector spaces. The determinant of a matrix gives a scalar, just as the integral of an integrable function gives a scalar. ...
2
votes
1answer
33 views

Matrices with the same characteristic polynomial

For all the $n \times n$ matrices, let's define an equivalent relation that two matrices are in the relation iff they have the same characteristic polynomial. How can we characterize the matrices ...
-1
votes
2answers
105 views

Show that exponential map is surjective [on hold]

How I can show that $\exp\colon \mathcal M(n,\mathbb C) \rightarrow \text{Gl}(n,\mathbb C)$ is surjective? Thank you.
12
votes
1answer
104 views

Check membership in a matrix group

I'm looking for a (preferably somewhat efficient) algorithm for this problem: Given a normal subgroup of $SL(m, \mathbb{Z})$ generated by a finite set $\{M_1, M_2, \dotsc, M_n\}$, and some $A \in ...
0
votes
2answers
28 views

Show determinant equals 0

Ok, i've been working on the following problem and this is what I've gotten: Let $F$ be a field, let $n$ be a positive integer, and let $A,B \in M{nxn} (F)$ be matrices satisfying $B\ne 0$ and ...
0
votes
0answers
16 views

Why the similarity transformation matrix in jordan block decomposition can not be chosen unitary?

We know that any matrix can be transformed into its jordan form by similarity transformation. But why can't we choose a Matrix S with its inverse $S^-1$ as unitary matrices for non diagonilizable ...
6
votes
1answer
52 views

Sparsest matrix with full inverse

What is the sparsest matrix in $\mathbb R^{n,n}$ such that the inverse is full? I.e. I am looking for a matrix $A\in \mathbb R^{n,n}$ with as few non-zero entries as possible, such that $A^{-1}$ has ...
0
votes
0answers
78 views

Finding smaller matrix in bigger matrix

Given a bigger matrix of size R*C .Where each element of matrix is between [-20,20]. Now i need to find a smaller matrix of dimension H*W (H <= R, W <= C) in such a way that sum of squared ...
3
votes
3answers
72 views

Why not $SL_n (\mathbb R)$ in this exercise

I just solved the following exercise: Let $SL_2(\mathbb Z)$ denote the set of $2\times2$ matrices with integer entries and determinant $1$. Prove that $SL_2(\mathbb Z)$ is a subgroup of ...
2
votes
2answers
256 views

How to read this matrix notation

Excuse me for this basic question, but when reading some mathematic books I have encountered the following matrix: W = 2diag([1 1 0,01]) Could anybody explain to ...
2
votes
2answers
52 views

why can identity matrix sometimes be trivially determined by context?

For any matrix $A$, $A I = I A = A$ ($I$ is the identity matrix). If $A$ has $m\times n$ dimension, the first identity matrix $I$ that appears in the above equations should have $n\times n$ ...
0
votes
1answer
31 views

A seemingly counterintuitive result on active and passive transformations of vectors

Let $\mathbf{v}$ be an element of a vector space with Euclidean $R^3$ as the underlying set. Assume the standard Cartesian basis $\{\mathbf{e^{(1)}, e^{(2)}, e^{(3)}}\}$ on it. Let $\mathbf{v^* = R ...
1
vote
2answers
56 views

Write Matrix $A$ to $A = \sum_{i=1}^{3} \lambda_i P_i$

Let $A$ be a Matrix: $$A = \begin{pmatrix} 1 & 0 & 3i \\ 0 & -3 & 0 \\ -3i & 0 & 1 \end{pmatrix}$$ Now I want to write $A$ as $$A = \sum_{i=1}^{3} \lambda_i P_i$$ I ...
2
votes
2answers
52 views

How to bring $5x_1^2 - 26x_1x_2 + 5x_2^2 + 10x_1 - 26x_2 = 31$ to the form $\langle x',Ax' \rangle = 1$

How can I bring $$5x_1^2 - 26x_1x_2 + 5x_2^2 + 10x_1 - 26x_2 = 31$$ to the form $$\langle x',Ax' \rangle = 1$$ where $x' = \alpha x + \beta$ where $\alpha \in \mathbb{R}^+$ and $\beta \in ...
2
votes
0answers
23 views

Random Rotation of Points using Householder matrices

I have $N$ points in $D$ dimensions, were $D$ is big, for sure more than $100$. $N$ is also big. The goal is to produce an algorithm in my code, that will take as input this dataset and will give ...
1
vote
2answers
61 views

Show that $\langle x, Ax \rangle + \langle b, x \rangle = c$ can be transformed to $\langle x', Ax' \rangle = 1$

Let $A$ be a real, regular, symmetric $n \times n$ matrix, $b \in \mathbb{R}^n$ and $c \in \mathbb{R}$ How can I show that $$\langle x, Ax \rangle + \langle b, x \rangle = c$$ can be transformed by ...
5
votes
1answer
61 views

Some questions about the pseudoinverse of a matrix

For every mxn-matrix A with real entries, there exist a unique nxm-matrix B, also with real entries, such that $$ABA = A$$ $$BAB = B$$ $$AB = (AB)^T$$ $$BA = (BA)^T$$ B is called the pseudoinverse ...
0
votes
2answers
63 views

$A^2 = 0$ and a square matrix A

$A$ is a square matrix, and $A^2 = 0$. Prove that for every a, the matrix $I+aA$ is invertible. Well, $(I + aA)(I -aA) = I$ and $I - (aA)^2 = I$ and from here $(aA)^2 = 0$, and from here $A^2 = 0$, ...
15
votes
9answers
894 views

Why do determinants have their particular form?

I know that for a matrix $A$, if $\det(A)=0$ then the matrix does not have an inverse, and hence the associated system of equations does not have a unique solution. However, why do the determinant ...
1
vote
1answer
28 views

Maple: Inverse of Covariance Matrix

Good day, I would like to ask about the inverse of covariance matrix from the coding below (using maple): Anyone know why the inverse A can't be computed? Thank you for your help!
5
votes
2answers
72 views

$A$ is diagonalizable and $A^3 = A^2$

If $A$ is diagonalizable and $A^3 = A^2$. Is it necessary true that $A^2 = A$?
21
votes
2answers
1k views

Determinant of a non-square matrix

I wrote an answer to this question based on determinants, but subsequently deleted it because the OP is interested in non-square matrices, which effectively blocks the use of determinants and thereby ...
4
votes
1answer
107 views

When is $R \, A^{-1} \, R^t$ invertible?

In the context of a Gaussian model, I came across a matrix product $R \, A^{-1} \, R^t$ where $R$ is a $m \times n$ rectangular matrix and as implied $A$ is $n \times n$ and invertible. On which ...
0
votes
2answers
42 views

Question on Eigen values

Let $A$ be a square matrix and $A^*$ be its adjoint, show that the eigenvalues of matrices $AA^*$ and $A^*A$ are real. Further show that $\operatorname {trace}(AA^*)=\operatorname {trace}(A^*A)$.
0
votes
0answers
41 views

Why does distance lose meaning in high-dimensional space?

I'm working on an algorithm that clusters points in extremely high-dimensional space (thousands, if not more). However, I came across this wikipedia page: ...