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), ...

learn more… | top users | synonyms (2)

0
votes
6answers
3k views

Why does a matrix have determinant zero if one row is the sum of two other rows?

So basically here I am trying to understand why it is like that? Suppose Matrix $$ A = \left( \begin{array}{ccc} a & b & c \\ d & e & f \\ a+d & b+e & c+f \end{array} \right) ...
1
vote
1answer
62 views

Finding matrix $A$ knowing that $A^2 = B$

Let $B$ be the $3\times3$ matrix $$ \begin{pmatrix} 1&8&5\\ 0&9&5\\ 0&0&4 \end{pmatrix}. $$ How can I find a triangular matrix $A$ with positive diagonal entries such that ...
0
votes
2answers
75 views

if $rank{(A - \lambda I)^k} = rank{(B - \lambda I)^k}$ then $A$ is similar $B$

Let $A,B \in M_n(\mathbb{R}).$ Suppose for all $\lambda \in \sigma (A)$ and for all $k \geq 0,$ we have $\mathrm{rank}(A - \lambda I)^k = \mathrm{rank}(B - \lambda I)^k.$ Then why are $A$ and $B $ ...
0
votes
1answer
61 views

If each eigenvalueof $A$ is either $+1$ or $-1$ $ \Rightarrow$ $A$ is similar to ${A^{ - 1}}$

Let $A \in {M_n}$ is nonsingular and each eigenvalue of $A$ is either $+1$ or $-1$.Why $A$ is similar to ${A^{ - 1}}$?
1
vote
1answer
60 views

Jordan form of different matrices

Suppose you have a 4x4 matrix with the characteristic polynomial equal to the minimal polynomial $m_F(x)=C_F(x)=(x-3)^2(x+2)^2$. Find the Jordan form. Is this the correct solution? $$ ...
1
vote
1answer
31 views

A question about non-linear least square method…

I am trying to fit a set of points into a sine function, using nonlinear least square method. The final step to obtain the derivative of its parameters is given by the equation (8) of: ...
0
votes
1answer
47 views

How can we find if a matrix is full column rank

If $A$ is an $n*k$ matrix with complicated form of elements. How can I show this matrix is full column rank? By complicated form I mean there is no known form for the elements of $A$.
1
vote
2answers
64 views

How is it distinguished in matrix multiplication which is the vector and which is the matrix representing a linear transformation?

The terminology that is used everywhere when applying a matrix to a "vector" is considered is this: the matrix represents a linear transformation and there is a row or column vector. But a matrix can ...
0
votes
1answer
84 views

Reverse Order Laws of M-P pseudoinverse

When I was writing a literature survey on Moore-Penrose pseudoinverse (literatures like this one, and this one), I encountered with the following equality which was named as reverse order law: ...
2
votes
0answers
16 views

Cartan matrices: motivation and intuitive examples?

could anyone provide me with a sketch of the motivation that gave rise to Cartan matrices in abstract (homological) algebra, Lie algebrae and so on? Which was the trigger or the need for them? It ...
26
votes
9answers
2k views

What does it mean to represent a number in term of a $2\times2$ matrix?

Today my friend showed me that the imaginary number can be represented in term of a matrix $$i = \pmatrix{0&-1\\1&0}$$ This was very very confusing for me because I have never thought of it ...
0
votes
1answer
74 views

norms of row matrices

Let $x_1,\ldots,x_m,y_1,\ldots,y_m$ be $k\times k$ sqaure matrices and assume $\|x_j\|\leqslant\|y_j\|$ for all $j=1,\ldots,m$ (the norm in $B(\ell_2^k)$). Now define the block matrices $x,y\in ...
2
votes
2answers
1k views

Derivative with respect to a matrix

How do we start with the matrix differentiation of this kind of equation? $$ V = \big[ y_t - Cx_t \big]^T R^{-1} \big[y_t - Cx_t \big] $$ here $x_t$ and $y_t$ are vectors and $C$ and $R$ are ...
-1
votes
1answer
30 views

Is this true that $A$ and $A^*$ are unitary equivalent(or equivalent)? .

Let $A \in {M_n}$.Is this true that $A$ and $A^*$ are unitary equivalent(or equivalent)?
0
votes
1answer
56 views

If ${\sum\limits_{i = 1}^n {\left| {{\lambda _i}} \right|} ^2} = \sum\limits_{i = 1}^n {{\sigma _i}^2} \Rightarrow$A is normal matrix

Let $A \in {M_n}$ have eigenvalues ${\lambda _1}.....{\lambda _n}$ and singular values ${\sigma _1}.....{\sigma _n}$ and suppose ${\sum\limits_{i = 1}^n {\left| {{\lambda _i}} \right|} ^2} = ...
1
vote
1answer
29 views

finding matrix represention for linear transformation for field extension

need some clarification. given an extension field K over F with F-linear transformation, for $\alpha \in K$, $f_\alpha(k) = \alpha \cdot k$ i.e. multiplication on the left. I need to find the ...
2
votes
0answers
39 views

Constrained zero diagonal low rank approximation of a matrix with zero diagonal

Suppose that you have a $n\times n$ matrix $A$ that is symmetric and has zero diagonal, such as for example $$ A=\pmatrix{ 0 & 2 & 2\\ 2 & 0 & 1\\ 2 & 1 & 0}, $$ and you want ...
0
votes
2answers
28 views

proof a theorem in linear algebra

prove that if λ1 and λ2 are two distinct eigenvalues of a matrix A and λ1 , λ2 are corresponding eigenvectors, respectively, then α1 and α2 are linearly independent please help... thank you...
1
vote
0answers
49 views

Formula for powering a matrix not working for all matrices

I'm currently learning about matrices and was asked to show that this formula works for powers of $M$. $$M^n = nM-(n-1)I$$ Where $M$ is the matrix (show below), $n$ is the exponent an $I$ is the ...
5
votes
3answers
58 views

If $A$ is a square matrix and $Ax = b$ has a unique solution for some $b$, is $A$ necessarily invertible?

Let $A$ be a square matrix. Suppose that $A x = b$ has a unique solution for some $b$. Is $A$ necessarily invertible? I said no because the invertible matrix theorem states that $A x = b$ has a ...
1
vote
2answers
73 views

Why does $tr({A^*}A) = \sum\limits_{i = 1}^n {{\sigma _i}^2} $?

Let $A \in {M_n}$ have eigenvalues ${\lambda _1}.....{\lambda _n}$ and singular values ${\sigma _1}.....{\sigma _n}$. Why does $tr({A^*}A) = \sum\limits_{i = 1}^n {{\sigma _i}^2} $?
1
vote
1answer
59 views

Distribution of a matrix product $\mathbf{a}^{H}\mathbf{H}\mathbf{b}$

Could someone help prove the following: I have two independent random vectors $\mathbf{a} \in \mathbb{C}^{M \times 1}$ and $\mathbf{b}\in \mathbb{C}^{N \times 1}$. Both $\mathbf{a}$ and $\mathbf{b}$ ...
-1
votes
1answer
104 views

Image of the product of a Matrix and its transpose

If $A$ is an $n\times m$ matrix, is it necessarily true that $\text{im}(A)=\text{im}(AA^T)$ where $A^T$ is the transpose of $A$.
1
vote
0answers
20 views

Similarity of orthogonal matrices

Prove that for any $M$ in $SO(3)$, there is a matrix $P$ in $SO(3)$ and a real $\alpha$ such that $$PMP^{-1} = \left[ \begin{matrix} \cos\alpha & \sin\alpha & 0 \\ ...
0
votes
1answer
88 views

Eigen vectors for matrix with unknown constants?

I have the following matrix: $$\begin{bmatrix}\alpha&0&0\\\beta-\alpha&\beta&0\\1-\beta&1-\beta&1\end{bmatrix}$$ So far I have worked out the polynomial to be: ...
-1
votes
1answer
27 views

How to find the matrix of the transformation relative to the basis?

Let $T:P_2\to P_2$ be the linear operator defined by $$T(a+bx+cx^2)=(3a+2b+4c)+(2a+2c)x+(4a+2b+3c)x^2$$ Find the matrix of the transformation $T$ relative to the basis $B=\{1,x,x^2\}$.
3
votes
1answer
66 views

Comparing/contrasting hyperbolic and Euclidean geometry - or, on how ${\rm PSO}_2(\Bbb R)$ sits inside ${\rm PSL}_2(\Bbb R)$

I am studying hyperbolic geometry, in particular comparing and contrasting it with familiar Euclidean geometry. Let $\Bbb E$ be the Euclidean plane, and $G={\rm Iso}^+(\Bbb E)$ be the group of ...
3
votes
2answers
90 views

Show a matrix satisfying $A^2 − 8A + 15I = 0$ is diagonalisable.

A square matrix $A$ (of some size $n × n$) satisfies the condition $A^2 − 8A + 15I = 0$. Show that this matrix is similar to a diagonal matrix. I know that we must show that 5 and 3 are the ...
4
votes
1answer
64 views

Question about eigenvalue of Hermitian matrix

This is an eigenvalue problem I found. Let $A$ be an $n$-by-$n$ Hermitian complex matrix and $u$ is a vector in $C^n$ such that $u^*u=1$. Let $k=u^*Au$. Show that there exists an eigenvalue $r$ of ...
1
vote
3answers
44 views

How can I show that the dimension of span is . . .?

The Krylov subspace generated by $n$-by-$n$ matrix $A$ is defined by : $K_k(A,x)=span\{x,Ax,A^2x,...,A^{k-1}x\}$ How do I show its dimension is at most $k$? I only know that $dim(span (V))=rank ...
2
votes
1answer
40 views

Proving matrix exponent property [closed]

How can I prove the following equation. I have tried but i couldn't. $$\exp(A(t_2+t_1))=\exp(At_2)\cdot \exp(At_1)$$ $A$ is a matrix Will I use state-transition matrix or what ? Thank you...
0
votes
2answers
94 views

sum of two wishart matrices

Assuming $\mathbf{H}_1\in\mathbb{C}^{K\times M}(M>K),\mathbf{H}_2\in\mathbb{C}^{K\times M} $, the entries of $\mathbf{H}_1\text{ and }\mathbf{H}_2$ are all i.i.d. $\mathcal{CN}(0,1)$. I know that ...
0
votes
1answer
46 views

How to see that $A = A^{-1}$ and $A^2 = A$ as quick and easy as possible without computer aid

I'm wondering which are the quickest/easiest methods to identify that the following relations hold for any given matrix: $A = A^{-1}$ and $A^2 = A$ On a computer it's easy and quick to identify if ...
1
vote
1answer
118 views

Shortcut finding D when diagonalizing a matrix when encountered with a Householder reflection

P is given as P = $\left(\begin{array}{rrr} 1 & 1 & 1\\ 1 & 0 & -2\\ 1 & -1 & 1 \end{array}\right).$ It is known that P is invertible. I is a 3x3 identity matrix Supposed ...
-1
votes
1answer
35 views

Simple question in Rank [closed]

Let $A,U \in {M_n}$ and $U$ is unitary matrix.Is this true that $Rank(AU)=Rank(A)$?
0
votes
4answers
50 views

Solving for a vector $x$ given $Ax=b$

This is a dumb question I know. If I have matrix equation $Ax = b$ where $A$ is a square matrix and $x,b$ are vectors, and I know $A$ and $b$, I am solving for $x$. But multiplication is not ...
0
votes
4answers
39 views

Linear algebra, inverse of a matrix

Prove that if $A$ and $B$ are square matrices such that $AB = I$ then $B$ is invertible and $A$ is inverse of $B$. Basically can you help me prove the uniqueness of the inverse of matrix?
0
votes
1answer
51 views

Is there a good book on Circulant Matrices?

I've done a bit of searching on Amazon and found the 1979 book by Philip J. Davis, but the typeset looks very old (blocked system print style) and I don't really like that. Does anyone know if there ...
3
votes
2answers
450 views

How adjacency matrix shows that the graph have no cycles?

Let $G$ a directed graph and $A$ the corresponding adjacency matrix. Let denote the identity matrix with $I$. I've read in a wikipedia article, that the following statement is true. Question. Is it ...
0
votes
1answer
52 views
1
vote
0answers
606 views

How to find value of an unknown in matrix to make system of linear equations consistent

I'm currently stuck on this question relating to finding the unknown in a matrix so that the system of linear equations is consistent. I need to solve for $\lambda$. My first instinct is to try and ...
0
votes
0answers
24 views

A Rayleigh quotient-related eigenvalue problem

This is an eigenvalue problem I found. Let $A$ be an $n$-by-$n$ Hermitian complex matrix and $u$ is a vector in $C^n$ such that $u^*u=1$. Let $k=u^*Au$. Show that there exists an eigenvalue $r$ of ...
0
votes
0answers
126 views

the SVD (singular value decomposition) of an augmented matrix

Suppose we have a $4\times 3$ dimensional matrix $A$. Denote the SVD of $A$ by $USV^T$, where $U\in R^{4\times 3}, S\in R^{3\times 3}, V\in R^{3\times 3}$. Then, we construct a new matrix $B=[A;0]\in ...
0
votes
0answers
7 views

Singular vectors of matrix products

Consider a matrix $M = U V^T$, where $U,V$ are the singular vectors. Is there a way to relate the matrices $U,V$ to the singular vectors of the product $Z = AMB^T$ ?
4
votes
1answer
86 views

Linear decomposition of positive semi-definite matrices

It is true that the vector space of $n\times n$ Hermitian matrices is an $n^2-$dimensional real vector space and that one can find a basis for this space consisting exclusively of positive ...
1
vote
1answer
62 views

Orthogonal matrices

Ok i'll reformulate my question. The real thing I have to prove is that for any M in SO(3), there is a basis $e_1 , e_2, e_3$ and a real $\alpha$ such as: M$e_1$ = cos$\alpha$$e_1$ + sin$\alpha$$e_2$ ...
4
votes
2answers
91 views

Matrices and Combinatorics are a bad combination.

Let $\scr A$ be the set of all $n\times n$ symmetric matrices all of whose entries are either $0$ or $1$ and such that if $n$ is even, $n^2/2$ of these entries are $1$ and $n^2/2$ of them are $0$, and ...
0
votes
1answer
30 views

Prove that the columns of the similarity matrix of a diagonalization are the eigenvectors

I'm interested in eigendecomposition of a matrix. It is clear for me, that you can eigendecompose a matrix if and only if it is diagonalizable. I'm looking for a short proof for that statement, that ...
0
votes
0answers
80 views

How do I get the rotation between two rotationmatrices?

I am stuck on a little rotation problem. The problem: I have 2 rotation matrices $A$ and $B$. $A$ and $B$ are relative to the coordinate system O. $A$ and $B$ are Quaternion rotation matrices. I am ...
3
votes
1answer
40 views

Problem on symmetric matrices

Let $A$ be square non-singular matrix of order $n \geq 2$. If $A$ is symmetric, then $A^2$ is symmetric positive definite. If $A^2$ is symmetric positive definite, then $A$ is symmetric. I think I ...