- To use this tool, please enter row values separated by spaces.
- End each row by hitting the
Enterkey so these are recognized as individual entries.
- Be sure that the input is free from extra spaces and lines, particularly if instead of typing you are pasting your input.
- The tool inverts a square matrix using Gauss-Jordan Elimination.
- A matrix filled with zeroes is returned if the input matrix is non-invertible. This is used just as a crude signal.
- A non-invertible square matrix, also called singular or degenerate, is one whose determinant is zero.
- Researchers, teachers, students, or anyone working with matrices.
- Invert the following 2x2 matrix:
- Invert the following 3x3 matrix:
2 6 -14
-3 -5 9
2 8 -16
- Invert the following 4x4 matrix:
2 5 -3 -2
-2 -3 2 -5
1 3 -2 0
-1 -6 4 0
- Invert the following 5x5 matrix:
3 0 0 0 0
2 -6 0 0 0
17 14 2 0 0
22 -2 15 8 0
43 12 1 -1 5
- The following information was found online (Quora, 2013; StackExchange, 2013a; 2013b).
Let Ʃ be a covariance matrix and Ʃ-1 an inverse covariance matrix, commonly referred to as the precision matrix.
With Ʃ, one observes the unconditional correlation between a variable i, to a variable j by reading off the (i,j)-th index.
It may be the case that the two variables are correlated, but do not directly depend on each other, and another variable k explains their correlation. By computing Ʃ-1 we can examine if the variables are partially correlated and conditionally independent.
Ʃ-1 displays information about the partial correlations of variables. A partial correlation describes the correlation between variable i and j, once you condition on all other variables. If i and j are conditionally independent then the (i,j)-th element of Ʃ-1 will equal zero. If the data follows a multivariate normal then the converse is true, a zero element implies conditional independence.
In general, Ʃ-1 is a measure of how tightly clustered the variables are around the mean (diagonal elements) and the extend to which they do not co-vary with the other variables (non-diagonal elements). The higher the diagonal elements, the tighter the variables are clustered around the mean.
Elaborate on this and propose an example using PCA (Principal Component Analysis).
- Carey, G. (1998). Important Matrices for Multivariate Analysis.
- IFW (2016). Leibniz Institute for Solid State and Materials Research, Dresden. Institute for Metallic Materials - Invert a 6x6 Matrix.
- Kurtz, M. (1991). Handbook of Applied Mathematics for Engineers and Scientists. McGraw Hill.
- Quora (2013). What is the inverse covariance matrix?.
- Sandefur, J. T. (1990). Discrete Dynamical Systems, Theory and Applications; Oxford University Press; Chapter 6 Absorbing Markov Chains (1990).
- StackExchange (2013a). How to interpret an inverse covariance or precision matrix?.
- StackExchange (2013b). What does the inverse of covariance matrix say about data? (Intuitively).
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