# Let $Q_1$ be an $m×n$ matrix with orthonormal columns and $Q_2$ be an $n×p$ matrix with orthonormal columns

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

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