# Intuition About Principal Component Directions

I am trying to really get a deep understanding of PCA. From my understanding, a principal component is defined as $$\mathbf{z}_k = \phi_{1,k} \mathbf{x}_1 + \ldots + \phi_{p,k} \mathbf{x}_p = \mathbf{X} \boldsymbol{\phi}_k, \tag{1}$$ where $$\boldsymbol{\phi}_k = (\phi_{1,k}, \ldots, \phi_{p,k})$$ is a vector of scalars and $$\mathbf{x}_j$$ is the $$j^{\text{th}}$$ predictor. In other words, a principal component is a linear combination of the original predictors. The loading vectors $$\boldsymbol{\phi}_k$$ are chosen to maximize the varaince of the principal components, i.e. maximize $$\mathrm{Var}(\mathbf{X} \boldsymbol{\phi}_k)$$, and as a result, each loading vector is orthogonal, i.e. $$\langle \boldsymbol{\phi}_k , \boldsymbol{\phi}_{\ell} \rangle = 0$$ unless $$k = \ell$$. Also, during the optimization process, we constrain each loading vector to be of unit length, so $$\| \boldsymbol{\phi}_k \|_2 = 1$$ for all $$k$$.

If we want to write this more compactly, if the columns of a matrix $$\mathbf{Z}$$ are the principal components and the columns of $$\mathbf{\Phi}$$ are the loading vectors, we have $$\mathbf{Z} = \mathbf{X} \mathbf{\Phi}. \tag{2}$$ As a result of the two conditions above, the matrix $$\mathbf{\Phi}$$ is orthogonal, meaning $$\mathbf{\Phi}^{-1} = \mathbf{\Phi}^T$$. So multiplying both sides of $$(2)$$ by $$\mathbf{\Phi}^T$$ gives us $$\mathbf{X} = \mathbf{Z} \mathbf{\Phi}^T \tag{3}.$$ It is worth noting that in practice, $$(3)$$ is calculated using the singular value decomposition $$\mathbf{X} = \mathbf{U} \mathbf{D} \mathbf{V}^T$$, where $$\mathbf{Z} = \mathbf{U} \mathbf{D}$$ and $$\mathbf{\Phi} = \mathbf{V}$$.

Re-writing the two matrices on the right side of $$(3)$$ as $$\mathbf{Z} = (\mathbf{z}_1, \ldots, \mathbf{z}_p)$$ and $$\mathbf{\Phi}^T = (\boldsymbol{\phi}_1^T, \ldots, \boldsymbol{\phi}_p^T)^T$$, we get \begin{align} \mathbf{X} &= \begin{pmatrix} \mathbf{z}_1 & \cdots & \mathbf{z}_p \end{pmatrix} \begin{pmatrix} \boldsymbol{\phi}_1^T \\ \vdots \\ \boldsymbol{\phi}_p^T \end{pmatrix} \\ &= \mathbf{z}_1 \boldsymbol{\phi}_1^T + \ldots + \mathbf{z}_p \boldsymbol{\phi}_p^T \\ &= (\mathbf{X} \boldsymbol{\phi}_1) \boldsymbol{\phi}_1^T + \ldots + (\mathbf{X} \boldsymbol{\phi}_p) \boldsymbol{\phi}_p^T\tag{4} \\ &= \mathbf{X} \Big( \boldsymbol{\phi}_1 \boldsymbol{\phi}_1^T + \ldots + \boldsymbol{\phi}_p \boldsymbol{\phi}_p^T \Big). \end{align} From this, it has to be true that $$\Big( \boldsymbol{\phi}_1 \boldsymbol{\phi}_1^T + \ldots + \boldsymbol{\phi}_p \boldsymbol{\phi}_p^T \Big) = \mathbf{I}$$. Here is some more empirecal evidence to show that this is true, using the simple case of two predictors:

set.seed(100)
x1 = rnorm(2000); x2 = x1 + 0.5*rnorm(2000)
mat = matrix(c(x1, x2), ncol = 2)
matsvd = svd(mat)
D = diag(2); diag(D) = matsvd$$d score = matsvd$$u %*% D
load = matsvd$v load[,1] %*% t(load[,1]) + load[,2] %*% t(load[,2]) [,1] [,2] [1,] 1 0 [2,] 0 1 My problem is that I cannot come up with an intuitive reason as to why this is true, and I was wondering if anyone could provide one. Is there any significant meaning behind $$\boldsymbol{\phi}_k \boldsymbol{\phi}_k^T$$? (As answered by @gunes below, since $$\mathbf{\Phi}$$ is orthogonal and square, we have $$\mathbf{\Phi} \mathbf{\Phi}^T = \mathbf{I}$$). EDIT I would also like to know if my definitions are correct. I stated that $$\boldsymbol{\phi}_k$$ is the loading vector for the $$k^{\text{th}}$$ principal component, and so the matrix $$\mathbf{\Phi}$$ would be the loading matrix. I am getting this definition from section 10.2.1 of An Introduction to Statistical Learning. However, I have also seen (for example, here), loading vector defined as $$\boldsymbol{\phi}_k = d_k \boldsymbol{v}_k$$, i.e. the $$k^{\text{th}}$$ loading vector is the $$k^{\text{th}}$$ right singular vector scaled up by the $$k^{\text{th}}$$ singular value. So which definition is correct? • To your last section about definitions. Some people, texts and programs call "loadings" the eigenvector entries and some call it these entries scaled up by the corresponding eigen- (or singular) values. The second way is better for a number of reasons, including linquistic, and I would strongly recommend following it. (cont.) – ttnphns Jul 18 '19 at 8:34 • (cont.) And even in the current Wikipedia article on PCA, if you read it through, you'll find that one paragraph implies word "loadings" is applicable to both unit-scaled direction vectors and them eigenvector-scaled, and another section later defines "loadings" as the label for the second only. – ttnphns Jul 18 '19 at 8:35 • Ah okay, so it would be best to have$\mathbf{\Phi}$be the principal directions, and the loadings would be given by$\mathbf{D} \mathbf{\Phi}\$. The notational inconsistency is annoying, as it makes learning about something new more difficult than it needs to be! – Aiden Kenny Jul 19 '19 at 1:09

We know (and you also stated) that $$\mathbf{\Phi}$$ is an orthogonal matrix, i.e. $$\mathbf{\Phi}\mathbf{\Phi}^T=\mathbf{I}$$. If we open the LHS, we'll have $$\mathbf{\Phi}\mathbf{\Phi}^T=[\mathbf{\phi}_1 \ ...\ \mathbf{\phi}_p]\left[\begin{matrix}\phi_1^T \\ ...\\ \phi_p^T\end{matrix}\right]=\sum_{i=1}^p\phi_i\phi_i^T=\mathbf{I}$$