But it carries over into really any size square matrix.

Well, then this guy is going to be a square matrix.

And a square matrix for which there is no inverse, of which an inverse is undefined is called a singular matrix.

And if he's a square matrix, what else do we know about it?

If we can assume that these, this is a square matrix with an orthonormal set.

If we have a square matrix with linearly independent columns, it means that C is invertible.

And this right here, this is C inverse, which is equal to C transpose, because C is an invertible matrix, or a square matrix, with orthonormal columns.

And we also know, since it's a square matrix with linearly independent columns, that C is invertible.

It can work in one direction or another if this matrix is non square, but it won't work in both.

In the last couple of videos, we've seen that if we have some matrix C that is n by n. It's a square matrix, and is columns, column form and orthonormal set.

We've learned about matrix addition, matrix subtraction, matrix multiplication.

And this is going to be a square matrix, where this has to be n, and this has to be n.

The identity matrix times any other matrix is just that matrix.

This is where we need it to be a square matrix where all of its columns are mutually orthogonal and they're all normal.

This matrix that I chose for an example is a good matrix. A non singular matrix. An invertible matrix.

So an identity matrix is a matrix.

Let's just do some matrix matrix products.

And by the matrix, I mean literally the matrix, like the movie The Matrix.

matrix

Matrix

Matrix

We just multiplied this matrix times this matrix.

This is our definition of matrix matrix products.

Or when I multiply that matrix times the identity matrix, I get the matrix again.

So its entire matrix representation is going to be guy's matrix times this guy's matrix.

You can actually multiply one matrix by another matrix.

So the matrix representation of the entire composition is going to be this matrix times this matrix.

Matrix size

Amended Matrix

The Matrix

Matrix encoding

Matrix Widget

First matrix

Second matrix

Matrix A.

Let's take this matrix and multiply it times another matrix.

This matrix right here will become that matrix right there.

But of course, if I multiplied the inverse matrix times the identity matrix, I'll get the inverse matrix.

For example, the matrix A above is a 3 2 matrix.

So in general, the nullity of any matrix of any matrix

the matrix A, the matrix of coefficients, then there's a vector

This is our transformation matrix S. This is our matrix A.

And I'll say I have another matrix. Let's say matrix Y is identical to matrix X except for this row.

And then we have C inverse, but because C was a square matrix with orthonormal columns, we know that C inverse is the same thing is C transpose.

Z Matrix Editor