# Tagged Questions

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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### Two vector spaces with same dimension and same basis, are identical?

Let $V$ subspace of $W$ and both have same dimension and same basis. Then can we safely say that $V= W$ ? I believe yes. For example there may be an element $x \in V$ written as a linear combination ...
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### Find m so that $(m+1,1,1)$ , $(1,-m,-1)$ , $(m,1-m,2)$ are linearly dependent

I formed an augmented matrix $$\left(\begin{array}{ccc|c}m+1&1&m&0\\1&-m&1-m&0\\1&-1&2&0\end{array}\right)$$ I now that we do reduced row echelon form for the ...
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### Singular idempotent matrices

We know that if a square matrix $A$ is idempotent, then $$A^2 = A$$ If $A$ is non-singular, then the only possible matrix that is idempotent is $A=I$. But if $A$ is singular, then are there ...
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### Invertibility of $BA$

I'm having trouble with the following question which may seem simple but to me it's not Let $A \in \mathbb{R}^{5 \times 7}, B\in \mathbb{R}^{7 \times 5}$. Prove that $BA$ is not invertible. ...
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### How tight is this trace inequality?

I would like to know how tight the following trace inequalities are for real symmetric $A$ and real symmetric $B \succeq 0$ $$\mbox{trace} (AB) \leq \lambda_{\max} (A) \cdot \mbox{trace} (B)$$ or ...
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### Minimum eigenvalue of a sum of symmetric matrices

Let $\{v_i\}$ be some orthonormal basis in $\mathbb{R}^n$, and let $\{w_i\}$ be a set of positive weights such that $\sum_{i=1}^n w_i = 1$. I am interested in bounding the smallest eigenvalue of the ...
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### Operatornorm of $(\mathbb{R}^d, \|\cdot\|_1) \to (\mathbb{R}^d, \|\cdot\|_{\infty})$

Determine the operatornorm of the mapping $I:(\mathbb{R}^d, \|\cdot\|_1) \to (\mathbb{R}^d, \|\cdot\|_{\infty})$! Unfortunately I haven't many ideas for this task. I know that the definition of the ...
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### commutativity of log(I + A) and log( A−1) (matrix function)

I'm self-(re)learning linear algebra since the beginning of the summer, and i have a problem with the following exercice entitled additive logarithmic. If i'm right, we need to prove the ...
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### Convex set or not?

This question is related to my previous question Set of all positive definite matrices with off diagonal elements negative I know that the set of all positive definite matrices form a convex set. ...
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### nth product of sequential matrices

$\forall n \in \mathbb{N}$, let: $$P_n = \left( \begin{matrix} a & 1-a \\ b_n & 1-b_n \end{matrix} \right).$$ Whereby $\{b_n\}_{n \in \mathbb{N}}$ is a monotonically increasing sequence of ...
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### Order of $\mathrm{SL}(n,\mathbb{F}_p)$ (Constructive proof)

Most proofs of $$\vert ~\mathrm{GL}(n,\mathbb{F}_p) ~\vert = \prod_{k=0}^{n-1} (p^n-p^k)$$ I have seen so far, are done by counting the possibilities to build up invertible matrices i.e. counting ...
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### Find $B(B^{T}B)^{-1}B^{T}$.

To find: $$B(B^{T}B)^{-1}B^{T}$$ for $B=[0,1,-1]^T$ I have $$\begin{bmatrix} 0\\ 1\\ -1 \end{bmatrix} \left ([0,1,-1]\begin{bmatrix} 0\\ 1\\ -1 \end{bmatrix} \right )^{-1}[0,1,-1]$$ but ...
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### Taylor series of a square matrix

Let $A$ be a constant square matrix, $\Delta t$ is a scalar. I would imagine a taylor series of its exponential like this: $$e ^{A\,\Delta t} = I + \sum_{n=1}^\infty \frac{(A\,\Delta t)^n}{n!}$$ ...
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### Write summation of vector outer products into matrix form

My question is as follows: Given the weighted summation of vector outer products $\sum_i\sum_jh_{ij}{\bf v_i}{\bf u_j}^T$, where $h_{ij}$ is the weight, and ${\bf v_i,u_j}$ are column vectors, I was ...
I encountered during my reading to ridge regression that $$(X^TX+\lambda I)^{-1}X^TX = I-\lambda(X^TX+\lambda I)^{-1}$$ What mathematical manipulation has been done here? Thanks in advance