AX=0 for any column vector X and A is a square matrix, A must be zero matrix?

AA is a square matrix and A=ATA=A^T, suppose that AX=0AX=0 for any column vector XX, how to prove that AA must be a zero matrix?

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Hint: if e1e_1 is the column vector (10⋮0)\pmatrix{1 \\ 0 \\ \vdots \\ 0}, what is Ae1Ae_1?
– Bye_World
2 days ago

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2 Answers
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You don’t need the condition that A=ATA = A^T. If A∈Mn(F)A \in M_n(\mathbb{F}) is a square matrix such that AX=0AX = 0 for all X∈FnX \in \mathbb{F}^n then AeiAe_i is the ii-th column of AA which must be zero and so A=0A = 0.

Considering A:Rnâ†′RnA:\mathbb R^n→\mathbb R^n, AX=0AX=0 for all X\in \mathbb R^nX\in \mathbb R^n implies dim(E_0)=ndim(E_0)=n, where E_0E_0 is eigenspace corresponding to eigenvalue 00. Hence the result.