# Questions tagged [linear-algebra]

For questions on linear algebra, including vector spaces, linear transformations, systems of linear equations, spanning sets, bases, dimensions and vector subspaces.

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### Determinant of the matrix $I+A+A^2+ \cdots + A^{n-1}$.

Let $c_1, \ldots, c_n$ be real numbers and let $A = (a_{ij})_{n\times n}$ the matrix defined by $a_{ii+1} = c_i$ for each $i = 1,2, \ldots, n-1$, $a_{n1} = c_{n}$ and the other entries are zero. Show ...
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### Find a matrix Q that has Q(Q^T)=I but does not have orthonormal columns

My textbook says that if a matrix Q is square and has orthonormal columns then Q(Q^T)=I, but it does not say the opposite (that if Q(Q^T)=I then Q has orthonormal columns). Is there an example of such ...
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### Dual space isomorphism non-canonical choice example

In a lot of resources that I have read it is mentioned that the isomorphism between $V$ and $V^*$ is non-canonical, but I was never sure that I properly understood precisely what this means. I haven't ...
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### Is there any solution to a system of inequalities $Ax>b$, where $x$ is a column vector and $A$ is a symmetric positive matrix? [closed]

So I need to solve the inequality $Ax>b$ where $x$ is a column and $A$ a symmetric positive matrix (in that case).
1 vote
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### Independence of two functions

Assume that f is a continuous function over the interval $[a,b]$ and there exists $x \in [a,b]$ such that $f(x) \neq 0$. I want to show that $f(x)$ and $xf(x)$ are linearly independent on this ...
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### Existence of nested sequence of separable polynomials with roots in [0,1]

Question: Is therea sequence (ak), k∈N of non-zero real numbers such that, for all n ∈ N, the polynomial Pn is separable( split with simple roots) in [0, 1]? My thoughts : I guess we are invited to ...
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### Maximum value of the element of matrix $f(M)$

$M \in M_n(\mathbb {Q}), rk(I-M) = tr (I-M), rk(M) = tr(M)$. What is the maximum value of the element of matrix $f(M)$, where $f = x^5 - x^4 + x^3 - x^2$? I am not sure, but I think that eigenvalues ...
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### Negative eigenvalue in PSD sum of matrices

Suppose that $A+B$ is positive semidefinite. Then the sum of the two matrices has only nonnegative eigenvalues. I want to prove that, given $B$ is of rank $1$, $A$ has at most $1$ negative eigenvalue, ...
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### Characterizing the constancy of the row rank of a matrix function when the rank is everywhere less than the maximum

In the theory of public economics I have occasion to examine the behavior of the following "matrix function": \begin{equation*} Z(x) = \begin{pmatrix} 1 & 1 & \cdots & 1 \\ g_{...
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### Is matrix idempotent? [duplicate]

Let't say we have a matrix A such that $rank(A)=tr(A)$ and $rank(I-A)=tr(I-A)$. I tried to prove $A^2=A$ given these properties but I failed. Yet I strongly suspect that this matrix A could be ...
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I want to find the spectral condition number of this matrix, notice that this matrix is symmetric, hence the spectral condition number can be wriiten as $$\frac{\max_{1\leq i \leq n }|\lambda_{i}| }{\... • 621 1 vote 0 answers 24 views ### Proving the Basis of a Scaled Vector Set J If  E = \left\{\overrightarrow{e_1}, \overrightarrow{e_2}, \overrightarrow{e_3}\right\}  is a basis, prove that  F = \left\{\alpha \overrightarrow{e_1}, \beta \overrightarrow{e_2}, \gamma \... 1 vote 1 answer 23 views ### Proof of the Single Value Decomposition I am working through the a proof of the single-value decomposition, from Strang's 'Introduction to Linear Algebra, 4th edition'. I have included the proof as shown in the book at the end of this post. ... • 373 -1 votes 1 answer 27 views ### Doubt related to Projection matrix I came across Projection matrix recently.. If we consider matrix A with columns A1,A2,.....An forming basis for a subspace. Then the formula for projection matrix for projecting a vector onto column ... 0 votes 0 answers 23 views ### Solving systems of linear algebraic equations in singular case I want to solve this system: A \in \mathbb{R}^{n \times n}, \quad Ax = b, \quad det(A) = 0 Hence, there are either no solutions or there are infinitely many of them, then I want to find a pseudo-... 0 votes 0 answers 16 views ### Finding parameter for positive semi-definiteness in the sum of two quadratic forms here's a problem I've come across. It involves determining the values of a parameter t for which a quadratic form h is positive semi-definite. I've briefly discussed this with some friends, and ... 10 votes 1 answer 41 views ### Understanding Nash Equilibria in a Bimatrix Game I am currently studying game theory and I came across a problem involving a bimatrix game. The bimatrix is given by:$$ (A, B) = \begin{pmatrix} (4, 2) & (0, 0) \\ (0, 0) & (1, 3) \end{...
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Find a real system of linear equations $Ax = b$ where A is 2 columns and 4 rows matrix and all elements of A are not zero, $b \in \mathbb{R}^4$, $z = (2\ 3)^T$ is the approximation of $Ax = b$ using ...