# Questions tagged [linear-algebra]

Linear algebra is concerned with vector spaces of all dimensions and linear transformations between them, including systems of linear equations, bases, dimensions, subspaces, matrices, determinants, traces, eigenvalues and eigenvectors, diagonalization, Jordan forms, etc. For questions specifically concerning matrices, use the (matrices) tag. For questions specifically concerning matrix equations, use the (matrix-equations) tag.

82,523 questions
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### Finding the inverse of a linear transformation

Let $A:P_3 \to P_3$ be linear operator such that $$Ap(x)=\int_0^1p(x+t)dt$$ where $p \in P_3$. Find $A(e)$ if $(e)=\{1,x,x^2,x^3\}$ and $A^{-1}(2x-x^3)$ I just started learning about linear operators ...
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### Let $A \in \mathbb R^{5 \times 5}$. Prove that $A^{4} + I \neq O$

Let $A \in \mathbb R^{5 \times 5}$. Prove that $$A^{4} + I \neq O$$ I understand that I have to start from $A^{4}+I= 0$ and come up with a contradiction. Is it the Cayley Hamilton I have to use?
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### Intersection of subspaces of direct sum is only zero

If $U_1,U_2,\dots,U_m$ are subspaces of $V$. Then $U_1+\dots+U_m$ is a direct sum if and only if $U_1 \cap \dots \cap U_m={\vec{0}}$. Is this true or not?
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### Prove that the determinant is greater than $1$

Let $A$ be an $n \times n$ matrix whose diagonal entries are strictly positive and off-diagonal entries are negative. The sum of the entries on each column is $1$. Prove that $\det(A) > 1$. I ...
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### Spectral Radius $\leq$ min(1-norm, infinity norm)

How do I prove that the spectral radius of a matrix is less than or equal to the minimum of 1-norm and infinity norm of the matrix? i.e. $$\rho(A) \leq min(||A||_1, ||A||_{\infty})$$ I know the ...
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### If zero is an eigenvalue are dimensions lost?

This is likely a silly question so sorry in advance. However, I am wondering if I am right in thinking that if zero is an eigenvalue, then some dimension must be lost. My understanding is that ...
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### Why are eigenvectors important for Deep Learning applications?

I know it is quite of a trite question to ask about the importance of eigenvectors, but I do not understand how they can be relevant for Deep Learning and when we can use them. Any reference to the ...
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### Showing that a subset of a given space of matrices is a subspace

I've been given the following $V$: The set of $2 \times 2$ matrices with real entries $H := \{ A \in V : A \; \text{is symmetric} \}$ And I need to prove that $H$ is a subspace of $V$. So far, I ...
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### Constructing quaternions - proof that square of each imaginary unit is -1

During construction of vector space of quaternions over real numbers I encountered a problem that I can't quite put my finger on. For the context: Hamilton a multiplication in plane that keeps ...
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### Meaning of (viable) and (non viable) flow pattern?

I found all the unknown traffic flows for this network (check picture), but in the end, the question is asking to give (one viable flow pattern) and (one that is not viable) what does that mean? any ...
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### Why is operator norm defined the way it is?

Is there an intuition for the infimum definition of ${\| A \|}_{\mathrm{op}}$ without using a different, equivalent definition? I am referring to the definition, given an operator $A: W \rightarrow V$...
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### Proving the difference of two matrices is PSD

Claim: For $x\in \mathbb{R}^n$, we have $\operatorname{Diag}(x) - xx^T \succeq 0$ if and only if $x_i \geq 0 \ \forall i\in [n]$ and $\sum_{i} x_i \leq 1$. Where $\operatorname{Diag}$ denotes the ...
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### Maximizing $x^T A^T B \, x$ over the unit Euclidean sphere
Is there an algorithm to solve the QCQP $$\begin{array}{ll} \text{maximize} & x^T A^T B \, x\\ \text{subject to} & \|x\|_2 = 1\end{array}$$ when $A^T B$ is not necessarily symmetric? When \$A^...