## Matrix Inverter

- To use this tool, please enter row values separated by spaces.
- End each row by hitting the
`Enter`

key 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:

7 -13

9 2 - 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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