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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4
votes
3answers
43 views

$A$ is a symmetric postivie definite matrix. Prove that $A^k$ is also a positive deinite

Let $A\in M_n(\mathbb{R})$, a symmetric positive-definite matrix. Prove that for every $k\in\mathbb{N}$, $A^k$ is also positive definite. So since $A\in M_n(\mathbb{R})$ is symmetric and positive ...
2
votes
1answer
30 views

$\left\| A \right\| \le \varepsilon \Rightarrow \left\| {\mathop A\limits^{\_\_} } \right\| \le \varepsilon$

Suppose $A \in {C^{n \times n}}$ $\left\| A \right\| \le \varepsilon$ such that $\left\| . \right\|$ is matrix norm subordinate to the euclidean vector norm. Is this true that $\left\| {\mathop ...
1
vote
1answer
32 views

Minimizing variance subject to linear inequality

Let A be a $n \times n$ matrix. Where $A$ is a symmetric positive definite matrix. Let $b$ be a vector in $R^n$. $x$ is an unknown vector to be determined. I'm interested to find vector $x$ such ...
2
votes
1answer
13 views

Commutative Monoid - matrix set

Let $M$={$\begin{bmatrix} a & b & c \\ c & a & b \\ b & c & a \end{bmatrix}|a,b,c\in \mathbb{R}, a+b+c=0$}. The matrices in $M$ are a special kind of Toeplitz matrices ...
2
votes
2answers
895 views
+50

rotating a rectangle via a rotation matrix

I want to rotate a 2D rectangle using a rotation matrix. After the rotation, I want the points (x, y) of the rectangle to be: ...
0
votes
3answers
64 views

Given $A^2-4A+I=0$, show that $ A^3=15A-4I$

If have a question like this , can we using equation method or deduction method to answer the question?? Or we need to answer the question by substituting the matrix??
1
vote
1answer
33 views

Proof for $\mathbf{M}$ unitary if $||\mathbf{M}\mathbf{v}|| = ||\mathbf{v}||$

Let $\mathbf{M} \in \mathbb{F}^{n, n}$. Then $||\mathbf{M}\mathbf{v}|| = ||\mathbf{v}||$($\mathbf{v} \in \mathbb{F}^n$) implies that $\mathbf{M}$ is unitary. My question is, how to prove this ...
0
votes
1answer
20 views

Analytical result for element-wise vector division?

I have two vectors $$a=[a_1,a_2,...,a_n], b=[b_1,b_2,...,b_n]$$ Is it possible to express the result $$c=[a_1/b_1,a_2/b_2,...,a_n/b_n]$$ by some standard matrix operations such as matrix ...
2
votes
4answers
136 views

Do row operations change the column space of a matrix?

I know that (i) row operations do not change the row space (ii) column operations do not change the column space and (iii) row rank = column rank (but this is sort of unrelated, I think). But, ...
0
votes
1answer
51 views

Which matrices diagonalizes a diagonal matrix? [on hold]

I think the answer is the set of all diagonal matrices but I am not sure. Can anyone give the answer with a proof?
0
votes
1answer
22 views

how to calculate the projection of a vector onto a closed convex set? [duplicate]

suppose we have a vector $x \in \mathbb{R}^{n}$, and a closed convex set $C \in \mathbb{R}^{n}$. $C =\{x|Ax=b\}$ how to calculate the vector $y \in \mathbb{R}^{n}$, which is the projection of $x$ ...
1
vote
1answer
25 views

Proving space of skew-symmetric matrices is orthogonal complement of symmetric matrices

Problem: Prove that $\left\{ A \in \mathbb{R}^{n \times n} \mid A \text{ is symmetric}\right\}^{\bot} = \left\{ A \in \mathbb{R}^{n \times n} \mid A \ \text{is skew-symmetric}\right\}$ with $\langle ...
1
vote
2answers
36 views

Help With Finding A Basis

I came up to the following matrix: $$\begin{pmatrix} 3 & 1& 3& -4\\ 0 & 0 & 0& 0\\ 0 & 0 & 0& 0\\ 0 & 0 & 0& 0\\ \end{pmatrix}$$ I know that ...
3
votes
2answers
32 views

For any $A, B \in SL(2, F)$, does knowing $\operatorname{tr}A$, $\operatorname{tr}B$, and $\operatorname{tr}AB$ specify $A$ and $B$?

In title, $F$ denotes a field. Does knowing the trace of two matrices and their product specify those two matrices? Up to some equivalence, perhaps?
0
votes
1answer
12 views

Question about row operations and row-echelon form,

If I have a matrix, with, say, the first two columns consisting of all zeroes, then is the first entry of the third column, which is non-zero, my first pivot variable, so that when solving Ax=b, for ...
1
vote
1answer
26 views

Matrix Properties Problem

If $A\in M(n\times n;R)$ and $K= \dfrac {A+A^T}{2} $ and $L= \dfrac{A-A^T}{2}$. Prove: i) that $K$ and $-L$ are symmetric ii) that $K+L=A$ iii) that $K$ and $L$ are unique matrices with the properties ...
3
votes
2answers
68 views

Eigenvalues of the sum of two matrices: one diagonal and the other not.

I'm starting by a simple remark: if $A$ is a $n\times n$ matrix and $\{\lambda_1,\ldots,\lambda_k\}$ are its eigenvalues, then the eigenvalues of matrix $I+A$ (where $I$ is the identity matrix) are ...
7
votes
2answers
127 views

Can you determine from the minors if the presented module is free?

Motivation (you can ignore this part): A problem in Hartshorne (II.5.8c) asks to show that if we have a coherent sheaf $\mathscr{F}$ on a reduced noetherian scheme $X$, and the function ...
2
votes
1answer
112 views

Asymptotic behavior of the minimum eigenvalue of a certain Gram matrix with linear independence

Consider the density matrices with the following spectral decompositions: $$\rho=\lambda_1|\nu_1\rangle+\lambda_{2}|\nu_2\rangle$$ and $$\sigma=\gamma_1|\omega_1\rangle+\gamma_2|\omega_2\rangle$$ such ...
6
votes
2answers
206 views

Determinant of a Certain Block Structured Positive Definite Matrix

PLEASE FIND THE EDITED VERSION OF THIS QUESTION HERE: Asymptotic behavior of the minimum eigenvalue of a certain Gram matrix with linear independence I WILL ALSO PUT UP A BOUNTY FOR THE EDITED ...
3
votes
2answers
35 views

Eigenvalues of Matrix Product.

Is there a relationship between the eigenvalues of individual matrices and the eigenvalues of their product? What about the special case when one of these matrices is a diagonal (positive) matrix? I ...
0
votes
0answers
17 views

Notation sumation confusion

I am reading paper about additive schwarz preconditioner, where following notation is used in order to obtain matrix C $$C_i = \sum_k (I^k B^k (P^k u_i)R^k)$$ . My question is, what's dimension of ...
3
votes
1answer
66 views

A matrix as a point in $\mathbb{R}^{nm}$

I just had a really quick question to ask. I was reading a book on linear algebra and have just been trying to wrap my head around what exactly a matrix represents. At one point, the book said "In a ...
1
vote
0answers
13 views

reoder basis vectors to get 'more diagonal' representation of NxM matrix

I am trying to reorder the basis vectors of an NxM matrix in a way that leads to a representation with 'as much weight close to the diagonal as possible'. I would be happy, if instead of a matrix ...
1
vote
0answers
13 views

Inner product inequalities with a diagonal matrix defining the inner product

This question came about from analyzing symmetric positive definite bilinear form decompositions and trying to understand what conditions would ensure certain inequalities hold. Suppose we have 3 ...
0
votes
2answers
80 views

Possible to express the diagonal matrix $D=\tiny\left(\begin{matrix}1&0&0\\0&2&0\\0&0&3\end{matrix}\right)$ as function of $3\times3$ identity matrix?

Is possible to express the diagonal matrix $$D=\left(\begin{matrix} 1 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 3 \end{matrix}\right)$$ as function of $3\times3$ identity matrix? For ...
0
votes
1answer
24 views

Calculate the matrices $[{A{π \over 4}]}^v$ for all possible values of v

Calculate the matrices $[{A{π \over 4}]}^v$ for all possible values of $v$, when $A(\varphi)=\left(\begin{matrix} \cos\varphi &\sin\varphi\\ -\sin\varphi & \cos\varphi\end{matrix}\right)$ . ...
1
vote
1answer
29 views

Different formulas for matrix transformations

I am a bit confused about how to get a matrix in a new basis. On the one hand, we always use the multiplication by transformation matrix when we want to receive a matrix in a new basis: $A' = CA$, ...
0
votes
2answers
28 views

Equivalence of two different versions of “change of basis matrix”?

I have a question regarding basis change and the matrix that represents it. I understand the concept, though I've noticed a different formula/proof in different math books and I don't understand how ...
0
votes
2answers
308 views

Minimal polynomial of a diagonal matrix

How can I show that the minimal polynomial of a diagonal matrix is the product of the distinct linear factors $(A-\lambda_{j}I)$? In particular, if we have a repeated eigenvalue, why is it that we ...
3
votes
2answers
244 views

If two matrices have the same trace and determinant, do their powers have the same trace?

Let $A,B$ be two $2 \times 2$ matrices over some finite field $\mathbb{F}_q$, such that they have the same trace and determinant. Does this imply that tr $A^k$ = tr $B^k$ for any integer $k$? I've ...
2
votes
0answers
15 views

derivative of quadratic form with regard to inverse of lower-triangular matrix

I have a quadratic form of the form $Q(\Sigma; x, \mu) = (x-\mu)'\Sigma^{-1}(x-\mu)$ where $\Sigma$ is a positive-definite non-singular matrix with (modified) Cholesky decomposition $\Sigma = LDL'$ ...
-2
votes
0answers
20 views

A qustion in matrix polynomial [on hold]

Definitions: ${A_j},{\Delta _j} \in {C^{n \times n}},(j = 0,1,2....m)$ ${\rm{P(}}\lambda {\rm{) = }}{{\rm{A}}_m}{\lambda ^m} + .....{A_1}\lambda + {A_0}$ is a matrix polynomial, and $\lambda $ is ...
1
vote
1answer
23 views

How to denote a tensor in terms of matrices product?

How to write a tensor in terms of a product of matrices? For example, I have $a \times b$ matrix $F$, and I want to create a 3D $a \times a \times a$ tensor $T$, where $T_{i,j,k} = \sum_{m=1}^{b} ...
1
vote
1answer
29 views

How do the rows of a change of basis matrix form a basis for expressing columns?

I am reading this article on Principal Component Analysis (PCA) and in section III-B (page 3) it has strange definition I don't understand. In the toy example $\mathbf{X}$ is an $m \times n$ ...
1
vote
1answer
50 views

Find the matrix $A$ of the orthogonal projection onto the line spanned by the vector $v$

Let $V$ be the plane with equation $x_1 + 4x_2 + 2x_3 = 0$ in $\mathbb{R}^3$. Find the matrix $A$ of the orthogonal projection onto the line spanned by the vector $v = \begin{bmatrix} -12 \\ 4 ...
0
votes
0answers
36 views

Calculation of $trace(L^THL)$, L is lower triangular, H is symmetric.

I am working on a problem where I had to find the following expression: $$ l = Tr({P'HP})$$ I already modified my model formulation using cholesky decomposition for PSD matrices and came up with ...
2
votes
1answer
1k views

Minimizing L1 Regularization

I have given a high dimensional input $x \in \mathbb{R}^m$ where $m$ is a big number. Linear regression can be applied, but in generel it is expected, that a lot of these dimensions are actually ...
3
votes
3answers
765 views

Are there non-affine matrices?

Matrices are useful for proving statements like The ratio between the areas of a parallelogram and the quadrilateral formed by joining their midpoints is $2$. The ratio between the volumes ...
0
votes
0answers
21 views

SVD proof to If $A$ is of full rank, then $A^{*}A$ is of full rank

Provided $A$ is a full rank matrix $\in\mathbb{C^{m\times n}}$, then $A^{*}A$ is of full rank. Suppose $m\gt n$. There is a solution to this problem: solution link, and the top solution makes ...
1
vote
1answer
20 views

Ranks of matrices after multiplication by a nonsingular matrix

Consider an $n \times n_1$ matrix $A_1$ and an $n \times n_2$ matrix $A_2$ with the following properties: $\mathrm{Rank} (A_1)=n_1$, $\mathrm{Rank} (A_2)=n_2$, $\mathrm{Rank} (A_1 : A_2)=n_1+n_2$ ...
1
vote
1answer
30 views

Connection between $S$ and $A$ when $S^{-1} A S$ is a diagonal matrix

In diagonalizing a matrix $A$, we use a matrix $S$, which consists of eigenvectors of $A$. To compute $S$, we simply take each eigenvector of $A$ and write it as a linear combination of the standard ...
2
votes
2answers
45 views

Positive semi-definiteness of a matrix whose diagonal elements slightly differ from the sum of the absolute values of other elements in the row

I have a matrix which has the following form: $$ A= \begin{bmatrix} a+b-\varepsilon_1 & -a & -b \\ -a & a+c-\varepsilon_2 & -c \\ -b & -c & b+c-\varepsilon_3 \end{bmatrix} $$ ...
2
votes
1answer
40 views

Comparing $\text{tr}(A^{-1})$ and $\text{tr}(A(B+A)^{-2})$ for pd $A$ and psd $B$

Suppose that $A$ is positive definite and $B$ positive semidefinite, both with dimension $n\times n$. Is there some inequality between $$ \text{tr}(A^{-1})\quad\text{and}\quad\text{tr}(A(B+A)^{-2})? ...
0
votes
2answers
57 views

Is “basis times square matrix” a new basis?

Suppose we have a vector space $V = (K, +, \cdot)$. Let $B$ be a basis for $V$. Now we take an arbitrary square matrix $S \neq 0$. $BS$ is just a linear combination of $B$. Thus $BS$ should be a new ...
1
vote
3answers
51 views

The rows of an orthogonal matrix form an orthonormal basis

A matrix $A \in \operatorname{Mat}(n \times n, \Bbb R)$ is said to be orthogonal if its columns are orthonormal relative to the dot product on $\Bbb R^n$. By considering $A^TA$, show that ...
0
votes
0answers
22 views

Reading a basis for $\operatorname{Null}(A)$ and $\operatorname{Null}(A^T)$ from the reduced row-echelon form of $A$

I know that it is possible to read the basis for $\operatorname{Null}(A)$ and $\operatorname{Null}(A^T)$ by simply looking at the reduced row-echelon form (RREF) of the matrix $A$. I have only an ...
0
votes
0answers
16 views

Why the transformation matrix is composed by the transformation of the vector in the basis?

I have seen that to construct a transformation matrix which represents a linear transformation, we just need to apply the transformation to the vectors in our basis, and the transformation matrix is ...
0
votes
1answer
39 views
0
votes
0answers
101 views
+50

Solving a BTTB system by BCCB extension that is highly structured and fewer degree of freedom

Consider a BTTB system generated by a simple $3\times 3$ matrix, $$ Col_1 = \begin{bmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ ...