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Existence of eigenvalues for self-adjoint maps in finite-dimensional inner product spaces

score 2.6k · Answer 1 · 2026-04-12 16:02:15

2.6k

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Existence of eigenvalues for self-adjoint maps in finite-dimensional inner product spaces

Published on 12 Apr 2026 - 16:02

score 1k · Answer 2 · 2026-04-15 16:23:13

1k

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If $\{u_1,\ldots,u_n\}$ is a complete orthonormal set for the inner product induced by $G$, then $\sum_{j=1}^n u_ju_j^T=G^{-1}$

Published on 15 Apr 2026 - 16:23

score 117 · Answer 3 · 2026-04-17 11:47:21

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Closed linear subset of a Hilbert space

Published on 17 Apr 2026 - 11:47

score 210 · Answer 4 · 2026-04-15 08:13:58

210

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if $E^2=E$, then $\text{Im}\;E\subset\left(\ker E+(\ker E)^\perp\right)$?

Published on 15 Apr 2026 - 8:13

score 161 · Answer 5 · 2026-04-15 02:26:50

161

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Can we conclude that $\text{Im}\;E\subset\left(M+M^\perp\right)$?

Published on 15 Apr 2026 - 2:26

score 61.2k · Answer 6 · 2026-04-13 01:35:25

61.2k

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Why does cross product give a vector which is perpendicular to a plane

Published on 13 Apr 2026 - 1:35

score 196 · Answer 7 · 2026-04-16 06:39:54

196

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$\|x -y\|+\|y-z\|=\|x-z\|$ implies $y= a x + b z$ where $a +b =1$

Published on 16 Apr 2026 - 6:39

score 2.6k · Answer 8 · 2026-04-16 12:53:27

2.6k

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Orthonormalize the set of functions {1,x}

Published on 16 Apr 2026 - 12:53

score 95 · Answer 9 · 2026-04-15 13:08:55

95

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Relations between $\|x+a\|$ and $\|x-a\|$ in a normed linear space.

Published on 15 Apr 2026 - 13:08

score 24k · Answer 10 · 2026-04-12 18:24:21

24k

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Derivation of the polarization identities?

Published on 12 Apr 2026 - 18:24

score 11.3k · Answer 11 · 2026-04-09 01:37:14

11.3k

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How to interpret the adjoint?

Published on 09 Apr 2026 - 1:37

score 208 · Answer 12 · 2026-04-15 07:14:31

208

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Euclidean Metric and Convexity

Published on 15 Apr 2026 - 7:14

score 2k · Answer 13 · 2026-04-11 15:59:14

2k

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norm induced by inner product and triangle inequality

Published on 11 Apr 2026 - 15:59

score 93 · Answer 14 · 2026-04-15 19:27:57

93

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How can one prove the inequality $(|r_1s_1|+\cdots+|r_ns_n|)^2\leq(r_1^2+\cdots+r_n^2)(s_1^2+\cdots+s_n^2)$ in $\mathbb{R}^n$?

Published on 15 Apr 2026 - 19:27

score 82 · Answer 15 · 2026-04-14 11:49:29

82

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If $\langle v,s\rangle+\langle s,v\rangle\leq \langle s,s\rangle$ for all $s\in S$, why is $v\in S^\perp$?

Published on 14 Apr 2026 - 11:49

List Question

Existence of eigenvalues for self-adjoint maps in finite-dimensional inner product spaces

If $\{u_1,\ldots,u_n\}$ is a complete orthonormal set for the inner product induced by $G$, then $\sum_{j=1}^n u_ju_j^T=G^{-1}$

Closed linear subset of a Hilbert space

if $E^2=E$, then $\text{Im}\;E\subset\left(\ker E+(\ker E)^\perp\right)$?

Can we conclude that $\text{Im}\;E\subset\left(M+M^\perp\right)$?

Why does cross product give a vector which is perpendicular to a plane

$\|x -y\|+\|y-z\|=\|x-z\|$ implies $y= a x + b z$ where $a +b =1$

Orthonormalize the set of functions {1,x}

Relations between $\|x+a\|$ and $\|x-a\|$ in a normed linear space.

Derivation of the polarization identities?

How to interpret the adjoint?

Euclidean Metric and Convexity

norm induced by inner product and triangle inequality

How can one prove the inequality $(|r_1s_1|+\cdots+|r_ns_n|)^2\leq(r_1^2+\cdots+r_n^2)(s_1^2+\cdots+s_n^2)$ in $\mathbb{R}^n$?

If $\langle v,s\rangle+\langle s,v\rangle\leq \langle s,s\rangle$ for all $s\in S$, why is $v\in S^\perp$?

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