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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3
votes
0answers
15 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
16 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
votes
2answers
34 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 ...
0
votes
2answers
41 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 ...
0
votes
0answers
5 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
votes
2answers
30 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 $A^TC^TB^T$ = $(BCA)^T$?
0
votes
1answer
27 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
15 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?
0
votes
1answer
20 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
2answers
23 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
41 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
votes
2answers
21 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
43 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 ...
1
vote
1answer
19 views

Matrix representation induced by quotient space

someone can help me with this question, I know how to solve ker(A) but I don't know how to develop matrix representation. Thanks!!!!!
1
vote
3answers
55 views

Are the $2\times 2$ symmetric matrices a ring?

Ok so I am looking at Rings. I saw somewhere that the $2 \times 2$ symmetric matrices with entries in $\mathbb{R}$ is a ring. But if we look at matrix multiplication I am not convinced: If $ A = ...
0
votes
0answers
9 views

How to Simplify/Rewrite this Expression into a Generalized Eigenvalue Problem - via Similarity perhaps?

I have the following optimization problem: \begin{eqnarray} min~b' y' Z (Z' \Omega Z)^{-1} Z' y b \end{eqnarray} such that $b'b=1$. The matrices are $Z \in R^{n \times k}$, matrices $y \in R^{n \times ...
0
votes
0answers
30 views

Lower and Upper Triangular Matrices

$A$ is an $n\times n$ matrix and $L$ is an $n \times n$ nonsingular lower triangular matrix. How can I prove that if $LA$ is lower triangular, then $A$ is lower triangular? How can I do the same for ...
1
vote
0answers
20 views

Elementary Matrices, Replacing a row

Lets say I have a 4 x 4 Matrix and I want to replace one row of that matrix with a different row from the same matrix. I need a matrix that when multiplied to the original matrix achieves this. I ...
0
votes
1answer
40 views

A question about a notation

Let $A$ be a non-singular square matrix. Which of the following notations is correct? $${A^2}^{-1} \qquad \text{or} \qquad A^{-2}$$
0
votes
1answer
25 views

Orthogonally diagonalizing a matrix

Can anybody explain how to orthogonally diagonalize the following matrix: $$ \begin{pmatrix} 9 & \sqrt10 \\ \sqrt10 & 0 \\ \end{pmatrix} $$ Am I correct in ...
0
votes
1answer
20 views

A connection between a matrix norm and a related matrix's largest eigen-value

I have been asked to prove that for $A\in M_n(\mathbb{C})$, with $||A||:=\sup_{x\in\mathbb{C}^n,|x|=1}|Ax|$, $$||A||=\sqrt{\lambda}$$ where $\lambda$ is the eigen value of largest modulus of $A^*A$. ...
2
votes
0answers
25 views

Following problem on topology $(N.B.H.M - 2015)$

let $X = \{ f \in C[-5 , 5] : f(-5) = f(5) = 0 \}$ . Then Which of the following statement are true : (a) There exist $f \in X$ such that $f \equiv 2$ on $[-1 ,0 ]$ and $f \equiv 3$ on ...
1
vote
2answers
28 views

Similarity in two 2x2 Matrices and finding the S in A=SBS-1

I am doing something wrong here and I am not sure what. The object of the exercise is to find the S for similar matrices $A$ and $B$. $A=SBS^{-1}$ with $B=\begin{pmatrix}4& 1\\1& ...
2
votes
2answers
50 views

Matrix determinants related by a polynomial factor

Show that LHS = $$\begin{vmatrix}a_1+b_1t & a_2+b_2t & a_3+b_3t \\ a_1t+b_1 & a_2t+b_2 & a_3t+b_3 \\c_1 & c_2 & c_3 \\\end{vmatrix}$$ RHS = (1-t^2) ...
0
votes
1answer
14 views

When does the Singular Value Decomposition fail?

Does the singular value decomposition ever not work? The statement of the associated theorem, here from wikipedia: http://en.wikipedia.org/wiki/Singular_value_decomposition#Statement_of_the_theorem is ...
4
votes
3answers
119 views

Complex square matrices. Difficult proof.

$det(I+A\cdot\bar{A}) \ge 0$ Is it possible to prove the inequality is true for all complex square matrices $A$ where $I$ is the identity matrix and $\bar{A}$ is the complex conjugated matrix.
0
votes
0answers
25 views

Linear transformation of vector

I have computer graphics class and i had something like that on lecture: $$ \begin{bmatrix} \overrightarrow{b1} & \overrightarrow{b2} & \overrightarrow{b3} \end{bmatrix} \begin{bmatrix} c1\\ ...
1
vote
1answer
36 views

Determinant and matrix power

I was wondering if there is a relation between the determinant of a matrix and the determinant of its powers. I mean I am looking for something like $$ \det (A^k) = f(\det(A), k). $$ A few check I ...
0
votes
2answers
35 views

given- $BA+B^2=I-BA^{2}$ what can be said about A ,B matrices

let A,B be $n\times n$ matrices such tha $BA+B^2=I-BA^{2}$ where $I$ is the identity matrix.which of the following is true 1.$A$ is nonsingular 2.$B$ is nonsingular 3.$A+B$ is nonsingular 4.$AB$ ...
1
vote
0answers
52 views

How to reach Moore-Penrose pseudoinverse solution to minimize error function

Edit I'm trying to figure the derivation of the Moore-Penrose pseudoinverse for linear regression. The starting expression is the standard error function. I'm not quite sure how to expand on this ...
0
votes
1answer
31 views

How to determine the signs of the eigenvalues of a symmetric $3\times 3$ matrix?

This is a homework problem: Let $a,b,c$ be positive real numbers such that $b^2+c^2<a<1$. If $A=\begin{pmatrix} 1&b&c\\b&a&0\\c&0&1\end{pmatrix}$, then which of the ...
0
votes
2answers
19 views

How to find the corresponding matrix of a dot product over a polynomial ring to a specific basis

Let $V= \mathbb R[x]_{\leq 2}$ be the vector-space of real polynomials with degree $\leq 2$. We define a dot product on the $V$ as follows: $$\left<f,g \right> = \int_{0}^1f(x)g(x)dx.$$ ...
0
votes
1answer
28 views

What is a transformation that can't have shearing called?

What is a transformation called when it can have separate scaling for x and y, rotation, and translation, but it cannot have shearing or scaling AFTER rotation? Basically if this transformation is ...
0
votes
1answer
12 views

matrix transformation help [on hold]

i'm really unsure of how to tackle the following questions (further maths gcse): the transformation A is represented by the matrix $$A = \begin{pmatrix} 2 & 0 \\ -1 & 3 \end{pmatrix}$$ ...
0
votes
2answers
40 views

Inequality $\sqrt[4]{x^TA^{-2}x}\sqrt{x^TAx}\leq 1$ for symmetric positive definite matrices

Assume that $x\in \mathbb{R}^{n}$ is a unit vector and $A$ is a symmetric positive definite matrix. Prove that $$\sqrt[4]{x^TA^{-2}x}\sqrt{x^TAx}\leq 1.$$ Progress Since A is spd, it is ...
0
votes
2answers
14 views

Rank of a matrix from a 5 X 7 matrix with a basis of 3 vectors

The question in my book is as follows: If the subspace of all solutions of Ax=0 has a basis consisting of thee vectors and if A is a 5 x 7 matrix, what is the rank of A? Now i thought because ...
0
votes
1answer
21 views

Given $\det(A)$ and $\det(B)$, is my calculation of $\det(-2B^T B A)$ correct?

Suppose $A$ and $B$ are $3 \times 3$ matrices with $\det(A) = -2$ and $\det(B) = -1$. What is the determinant of $C = -2 B^T B A$? I know that $$\det(A^T) = \det(A) \qquad \det(AB) = \det(A) ...
1
vote
1answer
17 views

Prove that if $C$ is anti hermitian matrix then $\forall v\in \mathbb C^n \ : \ Re(\langle Cv, v \rangle)=0 $.

Suppose $C \in M_{n\times n}(\mathbb C)$ satisfies $C+C^* = 0$. Prove that $\forall v\in \mathbb C^n \ : \ Re(\langle Cv, v \rangle)=0 $. Here is what I was able to show so far: We know that $C$ ...
0
votes
0answers
5 views

Variation of linear matrix inequality

When reading "Convex optimization, S. Boyd" p.76, Example 3.4, it says The last condition is a linear matrix inequality (LMI) in $(x,Y,t)$. Therefore, epi($f$) is convex. I am confused about ...
1
vote
0answers
12 views

Rectangular Orthogonal Matrix

Consider a overcomplete matrix $D$ of dimension $m\times n$ where $n>m$. I want to know under what conditions i can say $D$ has orthogonal columns or rows. More specifically when $D$ will be close ...
2
votes
0answers
44 views

Does this family of special matrices have a name?

These are the bisymmetric matrices that are "pyramid" shaped as follows: $$f(14) =\begin{bmatrix}1&1&1& 1& 1& 1& 1& 1& 1& 1& 1& 1& 1& 1\\1& ...
0
votes
2answers
72 views

Can we find the inverse for a vector

Can we inverse a vector like we do with matrices, and why ? I didn't see in any linear algebra course such a concept of vector inverse and I was wondering if there is any such thing and if not, why.
0
votes
2answers
30 views

Generating a random binary matrix with fixed number of nonzeros

I want an algorithm (just the idea, not the actual code) to generate a random $n$ by $n$ matrix with binary entries, but with the condition that the number of nonzeros must be a fixed number $c$. Any ...
1
vote
1answer
17 views

Blockwise Symmetric Matrix Determinant

This question arises from another one of mine, but separate enough that I feel it deserves its own thread. Wikipedia says that $$det\begin{bmatrix}A&B\\B &A \end{bmatrix} = ...
2
votes
2answers
39 views

Properties of the matrix square root

In a paper I am reading, it is claimed that if $A, B \in \mathbb{R}^{n \times n}$ are positive definite, then $$ A^{1/2} (A^{−1/2} B A^{−1/2})^{1/2} A^{1/2} = A (A^{-1}B)^{1/2} $$ because of the ...
4
votes
3answers
210 views

All Two by Two Matrices Satisfy a Certain Property Problem

Show that if $A$, $B$ are $2 \times 2$ matrices over $\mathbb{R}$ then there exists a real number $\lambda$ so that $$ (AB-BA)^2 = \lambda I $$ I can do this problem using brute force (i.e. looking ...
1
vote
0answers
41 views

$\text{Ker}A=\text{span}(u) \implies A=mat_C\left( u\wedge . \right)$

i found this equality and i wonder how can i find the right term $$\dfrac{1}{2}\left(\begin{matrix}0&1&1 \\ -1&0&1\\ -1&-1&0 ...
2
votes
1answer
23 views

Commutativity of the square root of matrices

Let $A, B \in \mathbb{R}^{n \times n}$ two positive definite matrices such that $AB = BA$, that is $A$ commutes with $B$. It is easy to prove that $A^{1/2}$ commutes with $A$, indeed $AA^{1/2} = ...
1
vote
0answers
25 views

GMRES and Preconditioning

I am using GMRES to approximate the solution of a system of equations $Ax=b$, I am using a preconditioner $P$ to make GMRES converge faster. My question is how do I know if the preconditioner I am ...
2
votes
1answer
57 views

determinant of matrix $X$

Please hint me. ‎How ‎can I ‎calculate ‎determinant ‎of ‎matrix ‎‎$‎X‎$‎?‎ \begin{equation*}‎ ‎\mathbf{X}=\left(‎ \begin{array}{ccc}‎ A&B&‎\cdots&B\\‎ B&A&‎\cdots& B\\‎ \vdots ...