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 views

Reference for the proof of interlacing of eigenvalues of submatrices

If one has a $n \times n$ Hermitian matrix $A$ and one removes $k$ of the rows and their corresponding columns then the eigenvalues of the remnant interlace the eigenvalues of the full matrix. Can ...
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1answer
17 views

Given a square matrix where $a_{11}=c\neq 0$ and $a_{ij}=0$ otherwise, can we find a matrix B such that B and A+B have no common eigenvalues?

Given a matrix where $a_{11}=c\neq 0$ and $a_{ij}=0$ otherwise, can we find a matrix B such that B and A+B have no common eigenvalues? If instead the matrix had its nonzero entry component at ...
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0answers
19 views

How to determine if a set represents a line, plane or hyperplane?

How do you approach a question that gives you a set and asked to determine if it represents a line, plane or hyperplane? The Question: https://www.dropbox.com/s/0gscqur18kqg3ma/SpanningQuestion.PNG ...
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0answers
16 views

How do I find the common invariant subspaces of a span of matrices?

Let $G_1, \ldots, G_n$ be a set of $m\times m$ linearly-independent complex matrices. Let $\mathcal{G} = \operatorname{span}\left\{ G_1, \ldots , G_n\right\}$ be the vector space that spans the set ...
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1answer
22 views

Solve the following matrix equation $X'X=A$

I have square matrices $X$,$A$ and $X'X-A=0$. $A$ is given and is positive definite and I need to get matrix $X$. I know $X$ is not unique since $TX$ such that $T'T=I$ will satisfy. My problem is ...
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1answer
26 views

Prove the equality of two determinants. [on hold]

Matrix $A = (a_{ij}) \in M (n x n, Field)$, Matrix $B = ((-1)^{i+j}a_{ij})$ Proof that $det(A)=det(B)$? Thanks in advance.
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1answer
18 views

Triangularization of a matrix.

so I need to find an invertible matrix $P$ such that $P^{-1}AP$ is upper-triangular, where $$A = \begin{bmatrix} 4 & -1 \\ 9 & -2 \end{bmatrix}$$ So I found that the eigenvalue is $1$ which ...
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1answer
14 views

Linear algebra - projection matrix - inverse matrix

I am not sure how to prove this one: Let $A$ be a projection matrix so that $A^2=A$ and $A$ is not equal to zero. Find the inverse matrix of $I+cA$. Thanks.
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2answers
52 views

Can a matrix have eigenvalue with infinite multiplicity?

Suppose we have matrix of the form $$ A= \begin{bmatrix} a & -1 \\ 0 & a \\ \end{bmatrix} $$ and we would like to analyze its diagonalizability. By taking the ...
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1answer
21 views

AB = Identity matrix; matrices; determinants; proof

Let $M(n\times n, \mathbb Z)$ be the set of all $n\times n$- matrices with integer coefficients, and a matrix $A \in M$. Proof, that: There is exactly one matrix $B \in M(n\times n, \mathbb Z)$ with ...
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1answer
34 views

$\operatorname{rank}(A) = $max number of rows of submatrix $B$; Proof

I don't understand how to proof the following: The rank of a matrix $A \in M$ ($m \times n$, Field) equals the maximum number of rows of a square submatrix $B$ of $A$ with $\det (B) \neq 0$. The ...
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0answers
16 views

Operator norm of a matrix less than or equal to one

Do all matrices of operator norm $\leq 1$ have the sum of the absolute values of their rows $\leq 1$?
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3answers
167 views

To show two matrices are conjugate to each other

Given two matrices A and B $$ A = \begin{pmatrix} 1 & 0 & 0 \\ 1 & 3 & 0 \\ 1 & 2 & 1 \end{pmatrix}, \quad B = \begin{pmatrix} 3 & 0 & 4 \\ 0 & 1 & 0 \\ 0 ...
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2answers
22 views

Eigenvalues of a transpose multiplication

Say I have a matrix $\mathbf B \in \mathbb R^{m\times n}$. Is it correct to say that the eigenvalues of $\mathbf B^T\cdot\mathbf B$ are always positive?
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0answers
21 views

What does matrix decomposition really mean?

Any element of the symplectic group $\operatorname{Sp}(2n,\mathbb{R})$ can be decomposed using the Euler decomposition into the product of three matrices. \begin{equation} S = O\begin{pmatrix}D & ...
2
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0answers
33 views

Is $2^{xy}$ a positive definite kernel?

Is $2^{xy}$ a positive definite kernel on $\mathbb{N}$? i.e. for all $a_1, ..., a_n \in \mathbb{R}$, for all $x_1, ..., x_n \in \mathbb{N}$, $\sum_{i,j} a_i a_j 2^{x_ix_j}\geqslant 0$
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0answers
24 views

does a closed form solution exist for this equation?

I have a cost function $J$, which depends on a projection matrix $W$, which is unknown. When I get the partial derivative $\frac{\partial J}{\partial W}$ the equation is: $\frac{\partial J}{\partial ...
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0answers
17 views

Gauss-Jordan elimination in the form of (A|I)

So Gauss-Jordan elimination can be performed through the form of $(A|I)$ where $I$ is the identity matrix. We carry out row elementary operations as usual until the matrix becomes the form $(I|B)$, ...
1
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1answer
16 views

The inverse of the sum of two matrices in *Applied statistical decision theory *.

I am following Applied statistical decision theory [by] Raiffa, Howard. Which can be consulted online here. A theorem at the page linked states that if two matrices $A,B$ are non-singular and of ...
3
votes
3answers
139 views

“Orthogonal” Rectangular Matrix

Is it possible to have a matrix $\mathbf B \in \mathbb R^{m\times n}$ such that it satisfies: $$\mathbf B^T\cdot\mathbf B = \mathbf I_n$$ Where $\mathbf I_n$ is the $n\times n$ identity matrix. Or ...
2
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1answer
28 views

How to prove this result using Permutations? [on hold]

Let A be the set of all $3*3$ skew symmetric matrices whose entries are either -1, 0 or 1. If there are exactly 3 zeroes, three 1's and three (-1)'s, then prove that only 8 such matrices can exist.
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2answers
35 views

Change of Basis for $2\times2$ matrix

Suppose I have the matrix basis $\begin{bmatrix}1&0\\0&0\\\end{bmatrix}$ , $\begin{bmatrix}0&1\\0&0\\\end{bmatrix}$ , $\begin{bmatrix}0&0\\1&0\\\end{bmatrix}$, ...
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votes
1answer
34 views

how to creat a matlab program? [on hold]

How can I creat a tridiagonal matrix in matlab if the elements of matrix is again a matrix instead of a scalar.n how to solve them using crouts methods.I am new to matlab.
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1answer
28 views

Determinant of $\lambda I + A^TA$

What properties $\lambda I + A^TA$ have? I know that $A^T A$ is positive semi-definite, and symmetric. I want to show that the determinant of $\lambda I + A^TA$ decreases as $\lambda$ increases!
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1answer
15 views

Find the standard matrix of T given T is a linear transformation

$T:\mathbb{R}^2\to \mathbb{R}^2$ first performs a horizontal sheer that transforms $e_2$ into $e_2 + 2e_1$ (leaving $e_1$ unchanged) and then reflects points through the line $x_2 = -x_1.$ I am ...
1
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1answer
21 views

Quickest way to calculate $(I-M)^{-1}(I-M^{n+1})$

What is the quickest way, as computationally efficient, to calculate $(I-M)^{-1}(I-M^{n+1})$, where $M$ is a given $n \times n$ singular matrix of $0$'s and $1$'s? My experiments show that the matrix ...
0
votes
2answers
45 views

Let $A,B \in \mathcal{M}_{2k+1}(\mathbb{C})$ such that $AB=0$, Prove that $|(A+A^T)(B+B^T)|=0$

Let $A,B \in \mathcal{M}_{2k+1}(\mathbb{C})$ such that $AB=0$, prove that $\det[(A+A^T)(B+B^T)]=0\ \ $ with $ \ k\in \mathbb{N}$ I don't have ideas for this problem. Thanks !
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2answers
16 views

Implications on structure of $B$ when $rank(A-B) = s$ for a fixed $A$

Consider the case where $A \in \mathbb{R}^{n \times K}$ where $n > K$ and $\text{rank}(A) = K$. Suppose we know $$ \text{rank}(A - B) = s$$ for some $s < K$. What does this say about the ...
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0answers
17 views

How to prove that a matrix with specific property is invertible?

If we have a square matrix $$ M = \begin{pmatrix} a_{11} & a_{12} & \cdots & a_{1n} \\ a_{21} & a_{22} & \cdots & a_{2n} \\ \vdots & \vdots & \ddots & ...
2
votes
1answer
46 views

Eigenspaces and jordan normal form

I have a question here regarding the jordan normal form of two matrices where the eigenspace is one is contained in the other. Let $A,B$ be two $nxn$ matrices s.t $AB=BA$. I firstly proved that the ...
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0answers
28 views

Sparse Matrices and Tridiagonalization.

Assume that we are given a sparse matrix,let it be 90*90(1000*1000), would you say that a vector with lots of zeros(let it be 90*1(1*1000),and 65(500) zeros are there),is a smart option to initialize ...
2
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0answers
31 views

When does $\| \Pi \|_1 = 1$ where $\Pi$ is a projection.

By projection I mean any matrix such that $\Pi = \Pi^2$. It is well known that all projections can be written as $\Pi = A(B^\top A)^{-1}B^\top$ for some $A,B$. Characterize the class of projections ...
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2answers
26 views

Show that if $v\in (V_c)^{\perp}$ then $(Av)\in (V_c)^{\perp}$ for a normal matrix $A$ with an eigenvalue $c$

Suppose $A \in M_{n\times n}(\mathbb C)$ is a normal matrix and $c$ is an eigenvalue of $A$. I'm trying to show that if $v\in (V_c)^{\perp}$ then $(Av)\in (V_c)^{\perp}$. I know that if we were ...
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1answer
40 views

Matrix representation of shape operator

Let $f$ be a parametrized surface $f: \Omega \subset \mathbb{R}^2 \rightarrow \mathbb{R}^3$ and $N : \Omega \rightarrow Tf$ the Gauß map. Then the shape operator is defined as $L = -DN \circ Df^{-1}.$ ...
3
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2answers
113 views

How to show that $A^2=AB+BA$ implies $\det(AB-BA)=0$ for $3\times3$ matrices?

Let $A$ and $B$ be two $3\times 3$ matrices with complex entries,such that $A^2=AB+BA$. Prove that $\det(AB-BA)=0$. (Is the above result true for matrices with real entries?)
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1answer
20 views

Clarification of some doubts on the definition of submatrix

I don't fully understand how I can choose a submatrix in a matrix. Judging from this definition and picture (http://mathworld.wolfram.com/Submatrix.html), I would assume that you can't pick as a ...
2
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1answer
50 views

Find the value of the Determinant

If $a^2+b^2+c^2+ab+bc+ca \le 0\quad \forall a, b, c\in\mathbb{R}$, then find the value of the determinant $$ \begin{vmatrix} (a+b+2)^2 & a^2+b^2 & 1 \\ 1 & (b+c+2)^2 ...
3
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1answer
42 views

Reversed Cayley transformation for any unitary matrix

It is well known that if $Q$ is a complex unitary matrix such that $I+Q$ is invertible (where $I$ is the identity matrix), that is, $-1$ is not an eigenvalue of $Q$, then $$ A:=(I-Q)(I+Q)^{-1} $$ is ...
0
votes
1answer
21 views

Proving that a matrix product is singular

I just played around in mathematica and found out that it seems like if $A$ is an $m\times n$ matrix and B is an $n\times m$ matrix, with $m>n$, then $AB$ is singular. How does one go about proving ...
3
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3answers
54 views

Rank in row echelon form

$$A= \begin{bmatrix} a & 1 & a & 0 & 0 & 0 \\ 0 & b & 1 & b & 0 & 0 \\ 0 & 0 & c & 1 & c & 0 \\ 0 & 0 & 0 & d & 1 ...
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2answers
43 views

Matrices that commute with all matrices [duplicate]

Let $Z_n$ be the set of all $n \times n$ matrices that commute with all $n \times n $ matrices. Show that $$Z_n = \{\lambda I_n \ | \ \lambda \in \mathbb R\}$$ ($I_n$ is the $n \times n$ identity ...
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0answers
10 views

Tangent line to the unitary group $U_1$

I have been working through a small project and the last part has me completely stumped. I have just shown that the matrix $\exp\tau X$ is unitary for all $\tau\in\mathbb{R}$ iff $X$ is ...
0
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2answers
33 views

Since $A^T B^T = (BA)^T$, what is $A^TC^TB^T$

It is well known $A^T B^T = (BA)^T$ So what would be the transpose of $A^TC^TB^T$? What conditions on C would make it true that $A^TC^TB^T$ = $(BCA)^T$?
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1answer
31 views

Property of orthogonal and skew symmetric matrix

If $A$ be a $n\times n$ orthogonal matrix and $X$ be a matrix such that $X=(A+I)^{-1}(A-I)$ then show that $X$ is a skew-symmetric matrix,whenever $n$ is an odd integer.
1
vote
1answer
17 views

Where does the identity matrix comes from in $(\alpha I - A) \vec X = 0$

Let $\alpha$ be a number, A a matrix, then $\alpha \vec X = A \vec X$ becomes $(\alpha I - A) \vec X = 0$ where does the I come from?
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1answer
22 views

Do we write a metric tensor as a matrix?

The metric tensor is an (0,2) tensor that is denoted by $g_{\mu\nu}$ in general relativity. I often see people write the metric field in matrix form like \begin{equation} g_{\mu\nu} = ...
2
votes
3answers
38 views

Should a reflection matrix of a vector have the same form as a rotation matrix?

According to the book: I know that it is not possible to write a reflection as a rotation, but from the text it seems that the matrix of the form $$ A=\left[ \begin{array}{ c c } a & ...
1
vote
3answers
44 views

Determine a matrix knowing its eigenvalues and eigenvectors

I read through similar questions, but I couldn't find an answer to this: How do you determine the symmetric matrix A if you know: $\lambda_1 = 1, \ eigenvector_1 = \pmatrix{1& 0&-1}^T;$ ...
0
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2answers
24 views

Interchange rows in a matrix without using interchange operation

I'm sure that it's already out there somewhere in the abyss that is page 37 on google, so I apologize. I haven't been able to find it. Given some arbitrary matrix, how can two rows be interchanged ...
3
votes
3answers
46 views

How can we calculate the exponential form of a rotation matrix

Considering the rotation matrix: $$ A(\theta) = \left( \begin{array}{cc} \cos\space\theta & -\sin \space\theta \\ \sin \space\theta & \cos\space\theta \\ \end{array} \right) $$ How can I ...