The Karcher mean of \(n\) positive definite matrices \(A_{1},\dots,A_{n}\) is defined as the unique minimizer (provided it exists) of the sum of squares of the Riemannian trace metric distances to each of the \(A_{i}\), i.e., \[ \Lambda (A_{1},\dots,A_{n})=\arg \min_{X\in {\mathbb P}}\sum\limits_{i=1}^{n}\delta ^{2}(X,A_{i})~, \] where \ \({\mathbb P}\) is the convex cone of \(m\times m\) positive definite matrices. The principal goal of this article is to construct a particular family of matrix means, each with numerous desirable properties such as monotonicity, that converges to the Karcher mean and to show that these properties are preserved in the limit. The new family of matrix means is defined by \[ X=\frac{1}{n}\sum\limits_{i=1}^{n}X\#_{t}A_{i}~,~\tag{*} \] where \(A~\#_{t}B=A^{1/2}(A^{-1/2}B~A^{-1/2})^{t}A^{1/2}\) . The authors prove that for each \(t\in (0,1]\) Eq. (*) has a unique positive definite solution, denoted by \(P_{t}(A_{1},\dots,A_{n})\), and show that each of these matrix means arises as a unique fixed point if a strict contraction for the Thompson metric. It is shown that these power means vary continuously with \(t\) and satisfy analogues of basic properties of power means of positive real numbers. The authors prove that the Karcher mean is the limit of power means as \(t\rightarrow 0\). This implies a simple and non-probabilistic proof of monotonicity, joint concavity and other new properties of the Karcher mean recently established by \textit{R. Bhatia} and \textit{R.L. Karandikar} [``Monotonicity of the matrix geometric mean'', Math. Ann. 353, No. 4, 1453--1467 (2012; Zbl 1253.15047)], and a globally convergent method for obtaining the Karcher mean by taking the limit of \(X_{k}=P_{1/k}(A_{1},\dots,A_{n})\).