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.
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If I'm given a square matrix 'A'
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an eigenvector associated with that matrix is a vector
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so that when I multiply the vector by 'A'
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the effect of multiplication is a scaling
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of that eigenvector
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and the scaling value is called the eigenvalue
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So if I'm given a square matrix 'A'
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an eigenvector, we'll call it 'v'
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associated with that matrix
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is a vector so that, when I multiply on the left
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by the matrix 'A'
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the effect of that multiplication is a scaling
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of the eigenvector by this value 'lambda'
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which is the eigenvalue associated
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with that eigenvector
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So, one more time
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The whole idea behind eigenvalues and eigenvectors
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is that, given a matrix 'A' - a square matrix
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an eigenvector associated with that matrix
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is a vector such that when I multiply
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by 'A' on the left,
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the effect of the multiplication is a scaling
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of the vector by its corresponding eigenvalue
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The eigenvector 'v' by definition is not the zero vector
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Let's suppose we have a nice matrix
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Make things small in dimension here
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let's say 3 by 1 and 1 by 3
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Now I want to take that matrix and
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we'll define a vector
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this will turn out to be an eigenvector
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as we'll see in just a moment
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Let's say a vector (1,1)
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So what is the result of multiplication here?
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'A' on the left times 'v'
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So multiply the matrix [(3,1) (1,3)]
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times the vector(1,1)
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Notice this matrix multiplication is defined again
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a vector can be considered as something of a
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degenerate or one dimensional matrix
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So when I perform the matrix multiplication
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the dot product here (3,1) dotted with (1,1)
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results in 4
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(1,3) similarly dotted with (1,1)
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results in 4
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So what do we have here?
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We have Av, my matrix times this vector
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Well the action of that multiplication
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in a geometric sense
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is a scaling of the original vector 'v'
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by a value of 4
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So Av in other words here is equal to
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lambda, where lambda is 4, my scaling factor
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times the original matrix
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So the bottom line is, to be eigenvector,
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when I multiply on the left by my matrix
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the action of this multiplication
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results in a scaling of my eigenvector
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in this case by the value 4
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I'd like to shed a little more light
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on this idea of eigenvalues and eigenvectors
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now from a geometric perspective
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So we'll continue with the example
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where we called the matrix 'A'
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this 2 by 2 matrix, [(3,1) (1,3)]
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and as you'll recall from the earlier example
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One of the eigenvectors associated
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with that matrix
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We'll call that vector v_1
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is the vector (1,1)
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So I've drawn that now on the plane
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And then the resultant action
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of multiplying the vector v_1 on the left
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by the matrix aforementioned 'A' here
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was this stretching or scaling by a factor 4 here
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And that was the lambda associated with
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that eigenvector
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So we can think of this geometrically here
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'Av_1' results in 4 times v_1
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Okay, so there's a nice kind of geometric rendering
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of what it means to be an eigenvalue
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and an eigenvector for a matrix
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As I also said earlier
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this particular matrix, as is common,
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with lots of matrices
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has other eigenvectors associated with it
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And I'll show you how to compute some of those
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things in just a moment here
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But one of the other eigenvectors
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associated with that particular matrix
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We'll call that vector v_2
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is the vector as I've drawn here (-1,1)
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And the eigenvalue associated with that
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particular eigenvector is the value 2
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So in other words, lambda is 2
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for this particular eigenvector
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So the result then, of multiplying
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let's say the vector v_2 on the left
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by the matrix A
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is that I get this scaled version
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in other words twice times the original vector here