The ridge estimator $(\hat{\beta}_R)$, and the expected value, are defined as;
\begin{align} \hat{\beta}_R &= \left( X'X + kI \right)^{-1}X'y, \ k \geq 0 \\ \text{E}\left( \hat{\beta}_R \right) &= \left( X'X + kI \right)^{-1}X'X \beta \end{align}
where $X \in \mathbb{R}^{n \times k}$, $\beta \in \mathbb{R}^{k \times 1}$, $\beta_R \in \mathbb{R}^{k \times 1}$.
It is stated directly in the textbook "Introduction to Linear Regression Analysis" that the following is true;
\begin{equation} \text{Bias} \left( \hat{\beta}_R \right)^2 = \left( E( \hat{\beta}_R ) - \beta \right)^2 = k^2 \beta' \left( X'X + kI \right)^{-2}\beta \end{equation}
How can I show this? If I expand the middle term, I get the following
\begin{equation} \left( E( \hat{\beta}_R ) - \beta \right)^2 = E( \hat{\beta}_R )^2 - E( \hat{\beta}_R )\beta - \beta E( \hat{\beta}_R ) + \beta^2 \end{equation}
However, $E( \hat{\beta}_R ) \in \mathbb{R}^{k \times 1}$ and $ \beta \in \mathbb{R}^{k \times 1}$. It appears to me that there is a problem with the dimension. How do I resolve this or is the claim in the textbook false?
As defined, the "squared bias" has no meaning since the bias is a vector. One should consider instead then squared norm of the bias, namely $B^2=[E(\hat{\beta}_R) - \beta]'[E(\hat{\beta}_R) - \beta]$. We have $$E(\hat{\beta}_R) - \beta= (X'X + k I)^{-1}(X'X - (X'X+kI))\beta=-k (X'X + k I)^{-1}\beta.$$ Therefore, $$B^2 = k^2 \beta'(X'X + k I)^{-2}\beta.$$