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I'm trying to calculate the standard matrix for orthogonal projection using the formula for the matrix P=$A(A^T A)^ {-1} A^T$.

My approach has diverged from the supplied answer to this question: my_attempt supplied

It seems that I've done the working for the inner part ($A^T A$) correctly but when it comes to multiplying the outer parts I have a very different answer. I've produced a $1 \times 1$ matrix while the supplied answer has a $3 \times 3$ matrix.

I really just started at the left and multiplied the outer A by the inverse of the inner multiplication: $\frac{1}{21}$, then multiplied the resulting vector with the outer $A^T$ .

For some reason I can't get Wolfram Alpha to calculate the matrix product either, it stops after the 1st multiplication Wolfram Alpha Equation

Would anyone be able to let me know what my mistake has been, and how I can properly approach this type of question in the future?

Thanks in advance!

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You make a mistake when multiplying the column vector with the row vector. Remember that $$ \begin{bmatrix} a \\ b \end{bmatrix} \begin{bmatrix} c & d \end{bmatrix} = \begin{bmatrix} ac & ad \\ bc & bd \end{bmatrix} $$

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  • $\begingroup$ Thank you so much for your reply! I think I might a bit tired, but I'm still not sure why Wolfram is not generating that answer, even when I split up the equations into separate inputs. $\endgroup$
    – Scottmeup
    Commented Nov 2, 2017 at 14:56
  • $\begingroup$ You forget the parentheses around the last row vector. Now it is interpreted as a one dimensional array instead of a two dimensional matrix (of which the length of the first dimension is 1). So try {{2, -1, 4}} instead of {2, -1, 4}. $\endgroup$
    – EdG
    Commented Nov 2, 2017 at 14:59
  • $\begingroup$ Argh, thank you so much! That's got it, I'm ashamed to say how much time I spent trying to see the problem! $\endgroup$
    – Scottmeup
    Commented Nov 2, 2017 at 15:10
  • $\begingroup$ You are welcome. These mistakes are a good way to learn :) $\endgroup$
    – EdG
    Commented Nov 2, 2017 at 15:37

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