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<title>Eigenvalues and Eigenvectors</title>
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<h1>Eigenvectors and Eigenvalues</h1>
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<h2>Characteristic equation</h2>
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<p>A is an n by n matrix, x is A non-zero vector, and if the number λ exists such that Ax= λx has A non-trivial solution x, then λ is the eigenvalue of A, and x becomes the eigenvector corresponding to λ.</p>
<p>Whether u and v are eigenvectors of A？ </p>
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<img src="images/eigenvalue.png" height= "55px"><br/><br/>
<img src="images/eigenvalue2.png" height= "55px">
<img src="images/eigenvalue3.png" height= "60px"><br/><br/>
<img src="images/eigenvalue4.png" height= "140px">
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<p>So, u is the eigenvector for the eigenvalue minus 4, but Av is not A multiple of v so v is not the eigenvector for A.</p>
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<h2>Eigenspace</h2>
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<p>(A- λI) x= 0 has non-trivial solutions, and the set of all solutions is the null space of the matrix A- λI, called the eigenspace of A corresponding to. The eigenspace consists of the zero vector and all the eigenvectors corresponding to λ.</p>
<p><img src="images/eigenvalue5.png" height= "35px">, Are there eigenvectors corresponding to the eigenvalue 7? </p>
<p>If and only if Ax=7x has A non-trivial solution, that is, Ax−7x=0, or (A−7I) x=0.</p>
<p>To understand the homogeneous equation, calculate </p>
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<img src="images/eigenvalue6.png" height= "40px" >
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<p>The columns of A−7I are clearly linearly dependent, so Ax=7x has A non-trivial solution, so 7 is the eigenvalue of A.</p>
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