An important property of an orthogonal matrix $$A$$ is that $$A A^T = A^T A = I$$, which means that its inverse is simply its transpose: $$A^{-1} = A^T$$. This property makes the matrix inverse operation computationally cheap.

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A matrix is orthonormal matrix if and only if its rows are mutually orthonormal and its columns are mutually orthonormal as well.

Linear Algebra - orthogonal matrix

To get the transpose of a matrix, we should do a mirror symmetry operation on the matrix according to one of its specific diagonal line, called the main diagonal.  The gotten matrix should be denoted as $A^T$.

Transpose

An n-dimensional identity matrix $I_{n}$ is a matrix that has ones on the diagonal and zeros as the rest of the elements.
The inverse of a matrix A can be defined as $A^{-1}$ such that $A^{-1}$A=$I_{n}$
We can use matrix inverse to solve a $Ax=B$: $x=BA^{-1}$

Linear Algebra - Identity and Inverse Matrices

Goodfellow, I., Bengio, Y., & Courville, A. (2016). $\mathit{Deep \ Learning.}$ MIT Press. Retrieved from [www.deeplearningbook.org](https://www.deeplearningbook.org) 

Deep Learning

Orthogonal Matrix Inverse Property

Vectors $$x$$ and $$y$$ are orthogonal if their dot product is zero, meaning that $$x^Ty=0$$.

Orthogonal Vectors

The Moore-Penrose pseudoinverse is used similarly to an inverse, but for nonsquare matrices. It can find approximate values for systems with no exact solution. Often denoted as $$A^\dagger$$, given an $$m \times n$$ matrix $$A$$ and a system $$A \vec{x} = \vec{y}$$, the pseudoinverse of $$A$$ is the matrix $$B$$ such that $$\vec{x} = B \vec{y}$$. Formal definition: $$A^\dagger = \lim_{\alpha \to 0} (A^T A + \alpha I )^{-1} A^T$$.

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