{"id":245,"date":"2023-10-23T19:51:57","date_gmt":"2023-10-23T23:51:57","guid":{"rendered":"https:\/\/pressbooks.bccampus.ca\/langarasandbox\/?post_type=chapter&p=245"},"modified":"2023-10-25T17:17:25","modified_gmt":"2023-10-25T21:17:25","slug":"math-2362-symmetric-matrices-and-diagonalization-7-2","status":"publish","type":"chapter","link":"https:\/\/pressbooks.bccampus.ca\/langarasandbox\/chapter\/math-2362-symmetric-matrices-and-diagonalization-7-2\/","title":{"raw":"Math 2362 - Symmetric matrices and diagonalization (7.2)","rendered":"Math 2362 – Symmetric matrices and diagonalization (7.2)"},"content":{"raw":"

MATH 2362 - Symmetric matrices and diagonalization (7.2)<\/h1>\r\nExample (last time):\u00a0<\/strong>Diagonalize the symmetric matrix<\/span> <\/span><\/span>\r\n\r\nA<\/mi>=<\/mo>[<\/mo>2<\/mn><\/mtd>1<\/mn><\/mtd>1<\/mn><\/mtd><\/mtr>1<\/mn><\/mtd>2<\/mn><\/mtd>1<\/mn><\/mtd><\/mtr>1<\/mn><\/mtd>1<\/mn><\/mtd>2<\/mn><\/mtd><\/mtr><\/mtable>]<\/mo><\/mrow><\/mstyle>{\"version\":\"1.1\",\"math\":\"A=\\left[\\begin{array}{lll} 2 & 1 & 1 \\\\ 1 & 2 & 1 \\\\ 1 & 1 & 2 \\end{array}\\right]\"}<\/annotation><\/semantics><\/math>\r\n\r\nCharacteristic polynominal: f<\/mi>(<\/mo>\u03bb<\/mi>)<\/mo>=<\/mo>(<\/mo>\u03bb<\/mi>\u2212<\/mo>1<\/mn>)<\/mo>2<\/mn><\/mrow><\/msup>(<\/mo>\u03bb<\/mi>\u2212<\/mo>4<\/mn>)<\/mo><\/mstyle><\/semantics><\/math><\/span>\r\n\r\n\r\n<\/mi><\/mstyle>{\"version\":\"1.1\",\"math\":\"f(\\lambda)=(\\lambda-1)^{2}(\\lambda-4)\"}<\/annotation><\/semantics><\/math>\r\n\r\nEigenvectors: \u03bb<\/mi>=<\/mo>1<\/mn><\/mrow>\u2015<\/mo><\/munder>:<\/mo>v<\/mi>\u2192<\/mo><\/mover><\/mrow>1<\/mn><\/mrow><\/msub>=<\/mo>[<\/mo>\u2212<\/mo>1<\/mn><\/mtd><\/mtr>1<\/mn><\/mtd><\/mtr>0<\/mn><\/mtd><\/mtr><\/mtable>]<\/mo><\/mrow>,<\/mo>v<\/mi>\u2192<\/mo><\/mover><\/mrow>2<\/mn><\/mrow><\/msub>=<\/mo>[<\/mo>\u2212<\/mo>1<\/mn><\/mtd><\/mtr>0<\/mn><\/mtd><\/mtr>1<\/mn><\/mtd><\/mtr><\/mtable>]<\/mo><\/mrow><\/mstyle><\/semantics><\/math><\/span> (and their non-zero linear combinations)<\/span>\r\n\r\n\u03bb<\/mi>=<\/mo>4<\/mn><\/mrow>\u2015<\/mo><\/munder>:<\/mo>v<\/mi>\u2192<\/mo><\/mover><\/mrow>3<\/mn><\/mrow><\/msub>=<\/mo>[<\/mo>1<\/mn><\/mtd><\/mtr>1<\/mn><\/mtd><\/mtr>1<\/mn><\/mtd><\/mtr><\/mtable>]<\/mo><\/mrow><\/mstyle>{\"version\":\"1.1\",\"math\":\"\\underline{\\lambda=4}: \\vec{v}_{3}=\\left[\\begin{array}{l} 1 \\\\ 1 \\\\ 1 \\end{array}\\right]\"}<\/annotation><\/semantics><\/math> (and its no-zero multiples)<\/span>\r\n\r\nThen<\/span>\u00a0<\/span>P<\/mi>=<\/mo>[<\/mo>v<\/mi>\u2192<\/mo><\/mover><\/mrow>1<\/mn><\/mrow><\/msub><\/mtd>v<\/mi>\u2192<\/mo><\/mover><\/mrow>2<\/mn><\/mrow><\/msub><\/mtd>v<\/mi>\u2192<\/mo><\/mover><\/mrow>3<\/mn><\/mrow><\/msub><\/mtd><\/mtr><\/mtable>]<\/mo><\/mrow>=<\/mo>[<\/mo>\u2212<\/mo>1<\/mn><\/mtd>\u2212<\/mo>1<\/mn><\/mtd>1<\/mn><\/mtd><\/mtr>1<\/mn><\/mtd>0<\/mn><\/mtd>1<\/mn><\/mtd><\/mtr>0<\/mn><\/mtd>1<\/mn><\/mtd>1<\/mn><\/mtd><\/mtr><\/mtable>]<\/mo><\/mrow><\/mstyle>{\"version\":\"1.1\",\"math\":\"P=\\left[\\begin{array}{lll} \\vec{v}_{1} & \\vec{v}_{2} & \\vec{v}_{3} \\end{array}\\right]=\\left[\\begin{array}{ccc} -1 & -1 & 1 \\\\ 1 & 0 & 1 \\\\ 0 & 1 & 1 \\end{array}\\right]\"}<\/annotation><\/semantics><\/math> diagonalizes\u00a0A<\/em>, with\u00a0<\/span>P<\/mi>\u2212<\/mo>1<\/mn><\/mrow><\/msup>A<\/mi>P<\/mi>=<\/mo>D<\/mi>=<\/mo>[<\/mo>1<\/mn><\/mtd>0<\/mn><\/mtd>0<\/mn><\/mtd><\/mtr>0<\/mn><\/mtd>1<\/mn><\/mtd>0<\/mn><\/mtd><\/mtr>0<\/mn><\/mtd>0<\/mn><\/mtd>4<\/mn><\/mtd><\/mtr><\/mtable>]<\/mo><\/mrow><\/mstyle><\/mstyle>{\"version\":\"1.1\",\"math\":\"\\small P^{-1} A P=D=\\left[\\begin{array}{lll} 1 & 0 & 0 \\\\ 0 & 1 & 0 \\\\ 0 & 0 & 4 \\end{array}\\right]\"}<\/annotation><\/semantics><\/math>\r\n\r\n \r\n\r\n \r\n\r\nTheorem:\u00a0<\/span><\/span><\/strong>If\u00a0A<\/em> is symmetric, any two eigenvectors associated with distinct eigenvalues are orthogonal.<\/span><\/span>\r\n\r\nDefinition<\/b><\/span><\/span>:<\/span><\/span><\/strong> A square matrix\u00a0P<\/em> is called orthogonal is\u00a0P<\/em> is invertible and\u00a0<\/span><\/span>P<\/mi>\u2212<\/mo>1<\/mn><\/mrow><\/msup>=<\/mo>P<\/mi>T<\/mi><\/mrow><\/msup>.<\/mtext><\/mstyle><\/mstyle>{\"version\":\"1.1\",\"math\":\"\\small P^{-1}=P^{T} \\text {. }\"}<\/annotation><\/semantics><\/math>\r\n\r\n \r\n\r\nTheorem:<\/span><\/span><\/strong> A matrix\u00a0P\u00a0<\/em>is orthogonal if and only if the columns (and rows) of\u00a0P\u00a0<\/em>form an orthonormal set.<\/span><\/span>\r\n\r\n \r\n

MATH 2362 - Orthogonal Diagonalization (7.2)<\/span><\/span><\/p>\r\n

Defintion:<\/span><\/span><\/strong> A square matrix\u00a0A<\/em> is called\u00a0orthogonally diagonalizable<\/span> if there is an orthogonal matrix diagonalizing\u00a0A<\/em>, i.e. there is an orthogonal\u00a0P\u00a0<\/em>such that\u00a0<\/span><\/span>P<\/mi>\u2212<\/mo>1<\/mn><\/mrow><\/msup>A<\/mi>P<\/mi>=<\/mo>D<\/mi><\/mstyle><\/mstyle>{\"version\":\"1.1\",\"math\":\"\\small P^{-1} A P=D\"}<\/annotation><\/semantics><\/math>.<\/p>\r\n

Theorem:<\/span><\/span><\/strong> An\u00a0n\u00a0<\/em>x\u00a0n<\/em> matrix\u00a0A<\/em> is orthogonally diagonalizable if and only if\u00a0A<\/em> is symmetric.<\/span><\/span><\/p>\r\n

Proof:<\/span><\/span><\/strong><\/p>\r\n

\u27f8<\/mo><\/mstyle><\/mstyle>{\"version\":\"1.1\",\"math\":\"\\small \\Longleftarrow\"}<\/annotation><\/semantics><\/math><\/p>\r\nIdea of proof: if A\u00a0<\/em>is symmetric matrix, all its eigenspaces are full-dimensional (i.e. geometric multiplicity=algebraic multiplicity for every eigenvalue).\r\n

\u27f9<\/mo><\/mstyle><\/mstyle>{\"version\":\"1.1\",\"math\":\"\\small \\Longrightarrow\"}<\/annotation><\/semantics><\/math><\/p>\r\n \r\n

Corollary: Symmetric matrices are always diagonalizable.<\/span><\/span><\/p>\r\n

Example:<\/span><\/span><\/strong> Orthogonally diagonalize the symmetric matrix\u00a0<\/span><\/span>A<\/mi>=<\/mo>[<\/mo>2<\/mn><\/mtd>1<\/mn><\/mtd>1<\/mn><\/mtd><\/mtr>1<\/mn><\/mtd>2<\/mn><\/mtd>1<\/mn><\/mtd><\/mtr>1<\/mn><\/mtd>1<\/mn><\/mtd>2<\/mn><\/mtd><\/mtr><\/mtable>]<\/mo><\/mrow><\/mstyle><\/mstyle>{\"version\":\"1.1\",\"math\":\"\\small A=\\left[\\begin{array}{lll} 2 & 1 & 1 \\\\ 1 & 2 & 1 \\\\ 1 & 1 & 2 \\end{array}\\right]\"}<\/annotation><\/semantics><\/math><\/p>\r\n \r\n

Characteristic polynominal:\u00a0<\/span><\/span>f<\/mi>(<\/mo>\u03bb<\/mi>)<\/mo>=<\/mo>(<\/mo>\u03bb<\/mi>\u2212<\/mo>1<\/mn>)<\/mo>2<\/mn><\/mrow><\/msup>(<\/mo>\u03bb<\/mi>\u2212<\/mo>4<\/mn>)<\/mo><\/mstyle><\/mstyle>{\"version\":\"1.1\",\"math\":\"\\small f(\\lambda)=(\\lambda-1)^{2}(\\lambda-4)\"}<\/annotation><\/semantics><\/math><\/p>\r\n \r\n\r\nEigenvectors:<\/span>\r\n

\u03bb<\/mi>=<\/mo>1<\/mn><\/mrow>\u2015<\/mo><\/munder>:<\/mo>v<\/mi>\u2192<\/mo><\/mover><\/mrow>1<\/mn><\/mrow><\/msub>=<\/mo>[<\/mo>\u2212<\/mo>1<\/mn><\/mtd><\/mtr>1<\/mn><\/mtd><\/mtr>0<\/mn><\/mtd><\/mtr><\/mtable>]<\/mo><\/mrow>,<\/mo>v<\/mi>\u2192<\/mo><\/mover><\/mrow>2<\/mn><\/mrow><\/msub>=<\/mo>[<\/mo>\u2212<\/mo>1<\/mn><\/mtd><\/mtr>0<\/mn><\/mtd><\/mtr>1<\/mn><\/mtd><\/mtr><\/mtable>]<\/mo><\/mrow><\/mstyle>{\"version\":\"1.1\",\"math\":\"\\underline{\\lambda=1}: \\vec{v}_{1}=\\left[\\begin{array}{c} -1 \\\\ 1 \\\\ 0 \\end{array}\\right], \\vec{v}_{2}=\\left[\\begin{array}{c} -1 \\\\ 0 \\\\ 1 \\end{array}\\right]\"}<\/annotation><\/semantics><\/math> (and their non-zero linear combinations)<\/span><\/p>\r\n \r\n

\u03bb<\/mi>=<\/mo>4<\/mn><\/mrow>\u2015<\/mo><\/munder>:<\/mo>v<\/mi>\u2192<\/mo><\/mover><\/mrow>3<\/mn><\/mrow><\/msub>=<\/mo>[<\/mo>1<\/mn><\/mtd><\/mtr>1<\/mn><\/mtd><\/mtr>1<\/mn><\/mtd><\/mtr><\/mtable>]<\/mo><\/mrow><\/mstyle>{\"version\":\"1.1\",\"math\":\"\\underline{\\lambda=4}: \\vec{v}_{3}=\\left[\\begin{array}{l} 1 \\\\ 1 \\\\ 1 \\end{array}\\right]\"}<\/annotation><\/semantics><\/math> (and its no-zero multiples)<\/span><\/p>\r\nThen<\/span> <\/span>P<\/mi>=<\/mo>[<\/mo>v<\/mi>\u2192<\/mo><\/mover><\/mrow>1<\/mn><\/mrow><\/msub><\/mtd>v<\/mi>\u2192<\/mo><\/mover><\/mrow>2<\/mn><\/mrow><\/msub><\/mtd>v<\/mi>\u2192<\/mo><\/mover><\/mrow>3<\/mn><\/mrow><\/msub><\/mtd><\/mtr><\/mtable>]<\/mo><\/mrow>=<\/mo>[<\/mo>\u2212<\/mo>1<\/mn><\/mtd>\u2212<\/mo>1<\/mn><\/mtd>1<\/mn><\/mtd><\/mtr>1<\/mn><\/mtd>0<\/mn><\/mtd>1<\/mn><\/mtd><\/mtr>0<\/mn><\/mtd>1<\/mn><\/mtd>1<\/mn><\/mtd><\/mtr><\/mtable>]<\/mo><\/mrow><\/mstyle>{\"version\":\"1.1\",\"math\":\"P=\\left[\\begin{array}{lll} \\vec{v}_{1} & \\vec{v}_{2} & \\vec{v}_{3} \\end{array}\\right]=\\left[\\begin{array}{ccc} -1 & -1 & 1 \\\\ 1 & 0 & 1 \\\\ 0 & 1 & 1 \\end{array}\\right]\"}<\/annotation><\/semantics><\/math> diagonalizes\u00a0A<\/em>, with\u00a0 <\/span>P<\/mi>\u2212<\/mo>1<\/mn><\/mrow><\/msup>A<\/mi>P<\/mi>=<\/mo>D<\/mi>=<\/mo>[<\/mo>1<\/mn><\/mtd>0<\/mn><\/mtd>0<\/mn><\/mtd><\/mtr>0<\/mn><\/mtd>1<\/mn><\/mtd>0<\/mn><\/mtd><\/mtr>0<\/mn><\/mtd>0<\/mn><\/mtd>4<\/mn><\/mtd><\/mtr><\/mtable>]<\/mo><\/mrow><\/mstyle><\/mstyle>{\"version\":\"1.1\",\"math\":\"\\small P^{-1} A P=D=\\left[\\begin{array}{lll} 1 & 0 & 0 \\\\ 0 & 1 & 0 \\\\ 0 & 0 & 4 \\end{array}\\right]\"}<\/annotation><\/semantics><\/math>\r\n\r\n ","rendered":"

MATH 2362 – Symmetric matrices and diagonalization (7.2)<\/h1>\n

Example (last time):\u00a0<\/strong>Diagonalize the symmetric matrix<\/span> <\/span><\/span><\/p>\n

A<\/mi>=<\/mo>[<\/mo>2<\/mn><\/mtd>1<\/mn><\/mtd>1<\/mn><\/mtd><\/mtr>1<\/mn><\/mtd>2<\/mn><\/mtd>1<\/mn><\/mtd><\/mtr>1<\/mn><\/mtd>1<\/mn><\/mtd>2<\/mn><\/mtd><\/mtr><\/mtable>]<\/mo><\/mrow><\/mstyle>{“version”:”1.1″,”math”:”A=\\left[\\begin{array}{lll} 2 & 1 & 1 \\\\ 1 & 2 & 1 \\\\ 1 & 1 & 2 \\end{array}\\right]”}<\/annotation><\/semantics><\/math><\/p>\n

Characteristic polynominal: f<\/mi>(<\/mo>\u03bb<\/mi>)<\/mo>=<\/mo>(<\/mo>\u03bb<\/mi>\u2212<\/mo>1<\/mn>)<\/mo>2<\/mn><\/mrow><\/msup>(<\/mo>\u03bb<\/mi>\u2212<\/mo>4<\/mn>)<\/mo><\/mstyle><\/semantics><\/math><\/span><\/p>\n

\n<\/mi><\/mstyle>{“version”:”1.1″,”math”:”f(\\lambda)=(\\lambda-1)^{2}(\\lambda-4)”}<\/annotation><\/semantics><\/math><\/p>\n

Eigenvectors: \u03bb<\/mi>=<\/mo>1<\/mn><\/mrow>\u2015<\/mo><\/munder>:<\/mo>v<\/mi>\u2192<\/mo><\/mover><\/mrow>1<\/mn><\/mrow><\/msub>=<\/mo>[<\/mo>\u2212<\/mo>1<\/mn><\/mtd><\/mtr>1<\/mn><\/mtd><\/mtr>0<\/mn><\/mtd><\/mtr><\/mtable>]<\/mo><\/mrow>,<\/mo>v<\/mi>\u2192<\/mo><\/mover><\/mrow>2<\/mn><\/mrow><\/msub>=<\/mo>[<\/mo>\u2212<\/mo>1<\/mn><\/mtd><\/mtr>0<\/mn><\/mtd><\/mtr>1<\/mn><\/mtd><\/mtr><\/mtable>]<\/mo><\/mrow><\/mstyle><\/semantics><\/math><\/span> (and their non-zero linear combinations)<\/span><\/p>\n

\u03bb<\/mi>=<\/mo>4<\/mn><\/mrow>\u2015<\/mo><\/munder>:<\/mo>v<\/mi>\u2192<\/mo><\/mover><\/mrow>3<\/mn><\/mrow><\/msub>=<\/mo>[<\/mo>1<\/mn><\/mtd><\/mtr>1<\/mn><\/mtd><\/mtr>1<\/mn><\/mtd><\/mtr><\/mtable>]<\/mo><\/mrow><\/mstyle>{“version”:”1.1″,”math”:”\\underline{\\lambda=4}: \\vec{v}_{3}=\\left[\\begin{array}{l} 1 \\\\ 1 \\\\ 1 \\end{array}\\right]”}<\/annotation><\/semantics><\/math> (and its no-zero multiples)<\/span><\/p>\n

Then<\/span>\u00a0<\/span>P<\/mi>=<\/mo>[<\/mo>v<\/mi>\u2192<\/mo><\/mover><\/mrow>1<\/mn><\/mrow><\/msub><\/mtd>v<\/mi>\u2192<\/mo><\/mover><\/mrow>2<\/mn><\/mrow><\/msub><\/mtd>v<\/mi>\u2192<\/mo><\/mover><\/mrow>3<\/mn><\/mrow><\/msub><\/mtd><\/mtr><\/mtable>]<\/mo><\/mrow>=<\/mo>[<\/mo>\u2212<\/mo>1<\/mn><\/mtd>\u2212<\/mo>1<\/mn><\/mtd>1<\/mn><\/mtd><\/mtr>1<\/mn><\/mtd>0<\/mn><\/mtd>1<\/mn><\/mtd><\/mtr>0<\/mn><\/mtd>1<\/mn><\/mtd>1<\/mn><\/mtd><\/mtr><\/mtable>]<\/mo><\/mrow><\/mstyle>{“version”:”1.1″,”math”:”P=\\left[\\begin{array}{lll} \\vec{v}_{1} & \\vec{v}_{2} & \\vec{v}_{3} \\end{array}\\right]=\\left[\\begin{array}{ccc} -1 & -1 & 1 \\\\ 1 & 0 & 1 \\\\ 0 & 1 & 1 \\end{array}\\right]”}<\/annotation><\/semantics><\/math> diagonalizes\u00a0A<\/em>, with\u00a0<\/span>P<\/mi>\u2212<\/mo>1<\/mn><\/mrow><\/msup>A<\/mi>P<\/mi>=<\/mo>D<\/mi>=<\/mo>[<\/mo>1<\/mn><\/mtd>0<\/mn><\/mtd>0<\/mn><\/mtd><\/mtr>0<\/mn><\/mtd>1<\/mn><\/mtd>0<\/mn><\/mtd><\/mtr>0<\/mn><\/mtd>0<\/mn><\/mtd>4<\/mn><\/mtd><\/mtr><\/mtable>]<\/mo><\/mrow><\/mstyle><\/mstyle>{“version”:”1.1″,”math”:”\\small P^{-1} A P=D=\\left[\\begin{array}{lll} 1 & 0 & 0 \\\\ 0 & 1 & 0 \\\\ 0 & 0 & 4 \\end{array}\\right]”}<\/annotation><\/semantics><\/math><\/p>\n

 <\/p>\n

 <\/p>\n

Theorem:\u00a0<\/span><\/span><\/strong>If\u00a0A<\/em> is symmetric, any two eigenvectors associated with distinct eigenvalues are orthogonal.<\/span><\/span><\/p>\n

Definition<\/b><\/span><\/span>:<\/span><\/span><\/strong> A square matrix\u00a0P<\/em> is called orthogonal is\u00a0P<\/em> is invertible and\u00a0<\/span><\/span>P<\/mi>\u2212<\/mo>1<\/mn><\/mrow><\/msup>=<\/mo>P<\/mi>T<\/mi><\/mrow><\/msup>.<\/mtext><\/mstyle><\/mstyle>{“version”:”1.1″,”math”:”\\small P^{-1}=P^{T} \\text {. }”}<\/annotation><\/semantics><\/math><\/p>\n

 <\/p>\n

Theorem:<\/span><\/span><\/strong> A matrix\u00a0P\u00a0<\/em>is orthogonal if and only if the columns (and rows) of\u00a0P\u00a0<\/em>form an orthonormal set.<\/span><\/span><\/p>\n

 <\/p>\n

MATH 2362 – Orthogonal Diagonalization (7.2)<\/span><\/span><\/p>\n

Defintion:<\/span><\/span><\/strong> A square matrix\u00a0A<\/em> is called\u00a0orthogonally diagonalizable<\/span> if there is an orthogonal matrix diagonalizing\u00a0A<\/em>, i.e. there is an orthogonal\u00a0P\u00a0<\/em>such that\u00a0<\/span><\/span>P<\/mi>\u2212<\/mo>1<\/mn><\/mrow><\/msup>A<\/mi>P<\/mi>=<\/mo>D<\/mi><\/mstyle><\/mstyle>{“version”:”1.1″,”math”:”\\small P^{-1} A P=D”}<\/annotation><\/semantics><\/math>.<\/p>\n

Theorem:<\/span><\/span><\/strong> An\u00a0n\u00a0<\/em>x\u00a0n<\/em> matrix\u00a0A<\/em> is orthogonally diagonalizable if and only if\u00a0A<\/em> is symmetric.<\/span><\/span><\/p>\n

Proof:<\/span><\/span><\/strong><\/p>\n

\u27f8<\/mo><\/mstyle><\/mstyle>{“version”:”1.1″,”math”:”\\small \\Longleftarrow”}<\/annotation><\/semantics><\/math><\/p>\n

Idea of proof: if A\u00a0<\/em>is symmetric matrix, all its eigenspaces are full-dimensional (i.e. geometric multiplicity=algebraic multiplicity for every eigenvalue).<\/p>\n

\u27f9<\/mo><\/mstyle><\/mstyle>{“version”:”1.1″,”math”:”\\small \\Longrightarrow”}<\/annotation><\/semantics><\/math><\/p>\n

 <\/p>\n

Corollary: Symmetric matrices are always diagonalizable.<\/span><\/span><\/p>\n

Example:<\/span><\/span><\/strong> Orthogonally diagonalize the symmetric matrix\u00a0<\/span><\/span>A<\/mi>=<\/mo>[<\/mo>2<\/mn><\/mtd>1<\/mn><\/mtd>1<\/mn><\/mtd><\/mtr>1<\/mn><\/mtd>2<\/mn><\/mtd>1<\/mn><\/mtd><\/mtr>1<\/mn><\/mtd>1<\/mn><\/mtd>2<\/mn><\/mtd><\/mtr><\/mtable>]<\/mo><\/mrow><\/mstyle><\/mstyle>{“version”:”1.1″,”math”:”\\small A=\\left[\\begin{array}{lll} 2 & 1 & 1 \\\\ 1 & 2 & 1 \\\\ 1 & 1 & 2 \\end{array}\\right]”}<\/annotation><\/semantics><\/math><\/p>\n

 <\/p>\n

Characteristic polynominal:\u00a0<\/span><\/span>f<\/mi>(<\/mo>\u03bb<\/mi>)<\/mo>=<\/mo>(<\/mo>\u03bb<\/mi>\u2212<\/mo>1<\/mn>)<\/mo>2<\/mn><\/mrow><\/msup>(<\/mo>\u03bb<\/mi>\u2212<\/mo>4<\/mn>)<\/mo><\/mstyle><\/mstyle>{“version”:”1.1″,”math”:”\\small f(\\lambda)=(\\lambda-1)^{2}(\\lambda-4)”}<\/annotation><\/semantics><\/math><\/p>\n

 <\/p>\n

Eigenvectors:<\/span><\/p>\n

\u03bb<\/mi>=<\/mo>1<\/mn><\/mrow>\u2015<\/mo><\/munder>:<\/mo>v<\/mi>\u2192<\/mo><\/mover><\/mrow>1<\/mn><\/mrow><\/msub>=<\/mo>[<\/mo>\u2212<\/mo>1<\/mn><\/mtd><\/mtr>1<\/mn><\/mtd><\/mtr>0<\/mn><\/mtd><\/mtr><\/mtable>]<\/mo><\/mrow>,<\/mo>v<\/mi>\u2192<\/mo><\/mover><\/mrow>2<\/mn><\/mrow><\/msub>=<\/mo>[<\/mo>\u2212<\/mo>1<\/mn><\/mtd><\/mtr>0<\/mn><\/mtd><\/mtr>1<\/mn><\/mtd><\/mtr><\/mtable>]<\/mo><\/mrow><\/mstyle>{“version”:”1.1″,”math”:”\\underline{\\lambda=1}: \\vec{v}_{1}=\\left[\\begin{array}{c} -1 \\\\ 1 \\\\ 0 \\end{array}\\right], \\vec{v}_{2}=\\left[\\begin{array}{c} -1 \\\\ 0 \\\\ 1 \\end{array}\\right]”}<\/annotation><\/semantics><\/math> (and their non-zero linear combinations)<\/span><\/p>\n

 <\/p>\n

\u03bb<\/mi>=<\/mo>4<\/mn><\/mrow>\u2015<\/mo><\/munder>:<\/mo>v<\/mi>\u2192<\/mo><\/mover><\/mrow>3<\/mn><\/mrow><\/msub>=<\/mo>[<\/mo>1<\/mn><\/mtd><\/mtr>1<\/mn><\/mtd><\/mtr>1<\/mn><\/mtd><\/mtr><\/mtable>]<\/mo><\/mrow><\/mstyle>{“version”:”1.1″,”math”:”\\underline{\\lambda=4}: \\vec{v}_{3}=\\left[\\begin{array}{l} 1 \\\\ 1 \\\\ 1 \\end{array}\\right]”}<\/annotation><\/semantics><\/math> (and its no-zero multiples)<\/span><\/p>\n

Then<\/span> <\/span>P<\/mi>=<\/mo>[<\/mo>v<\/mi>\u2192<\/mo><\/mover><\/mrow>1<\/mn><\/mrow><\/msub><\/mtd>v<\/mi>\u2192<\/mo><\/mover><\/mrow>2<\/mn><\/mrow><\/msub><\/mtd>v<\/mi>\u2192<\/mo><\/mover><\/mrow>3<\/mn><\/mrow><\/msub><\/mtd><\/mtr><\/mtable>]<\/mo><\/mrow>=<\/mo>[<\/mo>\u2212<\/mo>1<\/mn><\/mtd>\u2212<\/mo>1<\/mn><\/mtd>1<\/mn><\/mtd><\/mtr>1<\/mn><\/mtd>0<\/mn><\/mtd>1<\/mn><\/mtd><\/mtr>0<\/mn><\/mtd>1<\/mn><\/mtd>1<\/mn><\/mtd><\/mtr><\/mtable>]<\/mo><\/mrow><\/mstyle>{“version”:”1.1″,”math”:”P=\\left[\\begin{array}{lll} \\vec{v}_{1} & \\vec{v}_{2} & \\vec{v}_{3} \\end{array}\\right]=\\left[\\begin{array}{ccc} -1 & -1 & 1 \\\\ 1 & 0 & 1 \\\\ 0 & 1 & 1 \\end{array}\\right]”}<\/annotation><\/semantics><\/math> diagonalizes\u00a0A<\/em>, with\u00a0 <\/span>P<\/mi>\u2212<\/mo>1<\/mn><\/mrow><\/msup>A<\/mi>P<\/mi>=<\/mo>D<\/mi>=<\/mo>[<\/mo>1<\/mn><\/mtd>0<\/mn><\/mtd>0<\/mn><\/mtd><\/mtr>0<\/mn><\/mtd>1<\/mn><\/mtd>0<\/mn><\/mtd><\/mtr>0<\/mn><\/mtd>0<\/mn><\/mtd>4<\/mn><\/mtd><\/mtr><\/mtable>]<\/mo><\/mrow><\/mstyle><\/mstyle>{“version”:”1.1″,”math”:”\\small P^{-1} A P=D=\\left[\\begin{array}{lll} 1 & 0 & 0 \\\\ 0 & 1 & 0 \\\\ 0 & 0 & 4 \\end{array}\\right]”}<\/annotation><\/semantics><\/math><\/p>\n

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