Let $Q_1$ be an $m×n$ matrix with orthonormal columns and $Q_2$ be an $n×p$ matrix with orthonormal columns. Show that $Q_1Q_2$ has orthonormal columns.

My proof.

Check if $Q^TQ = I$ for $Q_1Q_2$, check $(Q_1Q_2)^TQ_1Q_2 = Q_2^TQ_1^TQ_1Q_2 = Q_t^TIQ_2 = I$ so therefor I have proved my question

  • $\begingroup$ That is correct $\endgroup$ – Luiz Cordeiro Nov 24 '16 at 17:02
  • $\begingroup$ @Luiz Cordeiro I was not sure if that was for orthonormal or orthogonal, I am looking for orthonormal $\endgroup$ – bjp409 Nov 24 '16 at 17:05
  • $\begingroup$ You are applying the right criterion. A matrix $A$ is orthogonal if $A^T A$ is diagonal, and ortonormal if $A^TA=I$. $\endgroup$ – Luiz Cordeiro Nov 24 '16 at 17:16
  • $\begingroup$ okay thanks for your help $\endgroup$ – bjp409 Nov 24 '16 at 17:24

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