#Matrix definition how to#

More on diagonalization in later lessons. We are still a bit away from our lesson in diagonalization, but for now we can say that yes, a zero matrix is diagonalizable since its zero elements can easily contain linearly independent eigenvectors. All the entries being the same, the matrix would be exactly the same, there is not an 'inverse' or 'opposite' from that. But let us think about this idea for a minute: If we have mentioned before that for any matrix of a certain size or dimensions, there exists only one configuration in which all of its entries are zero, therefore, there cannot be other way in which you can rearrange the zeros to obtain an inverse matrix which would have the same dimensions. Again, we will talk a bit more about this on our next lessons about inverting matrices. There are a few rules which can prove this, such as its determinant being zero and if a square matrix, its rank smaller than its dimensions. For now, we will plainly say that a zero matrix is not invertible.

#Matrix definition how to#

If you think this idea more in depth, you will realize that any non zero matrix cannot have a rank smaller than one, in other words, in order for any matrix to have a rank of zero, it must contain all zero elements inside, and so, our conclusion is that only zero matrices have a rank of zero.ДЇor practical purposes we will leave the complete explanation on how to know if a matrix is invertible or not, and how to invert those which are for our next lessons talking about the 2x2 invertible matrix. Thus, if you think about it, a zero matrix contains zero amount of linearly independent columns, and zero amount of non-zero rows, and so our final conclusion is that the rank of zero matrix must be zero. Can you actually row reduce it into a row echelon form? No. Do they happen to represent any dimension plane then? No. How can we obtain this for a zero matrix then? For that we first need to ask ourselves, are the vectors inside a zero matrix linearly independent from one another? Not really, they are all the same, and they are all zero vectors. So, if we talk about a solvable system of linear equations transformed into a matrix notation, finding the rank of such matrix allows us to see the maximum amount of independent variables, and thus dimension planes for which the system could be graphically represented.

We can define the rank of a matrix by computing its row echelon form and then counting the left non-zero rows, the purpose of which is to find the dimension of the vector space for the matrix in question. Remember that the rank of a matrix corresponds to the maximum amount of linearly independent columns inside the matrix. Once we have learned the zero matrix definition, let us talk about a few of this matrix's special characteristics.

Thus, both of these two types of matrices are what we call a zero trace matrix. The similarity between a regular zero matrix and a hollow matrix comes from their trace (the addition of the elements on their diagonals) since both have all zero elements to be added to produce a trace equal to zero. Such zero diagonal matrix usually refers to a hollow matrix, where all the diagonal elements inside of it are zero, while the rest of its elements can be any number.

Thus, you can have a zero matrix with any amount of rows or columns, but remember, for any given size it is possible to obtain only one zero matrix (which makes sense, since there is only one way to have all zeros as entries in a matrix of a particular size or dimension combination).Дo not confuse the zero matrix with something people may call a 'zero diagonal matrix'. Equation 2: Examples of zero matrices in various sizesДЇrom the zero matrix notation examples above, notice that these matrices can come in any size and dimension combination, and they are not necessarily square matrices.